Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 14
Published $\text {1968}$
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$14 \quad$ Indefinite Integrals
In the following, $u, v, w$ are functions of $x$; $a, b, p, q, n$ any constants, restricted if indicated; $e = 2.71828 \ldots$ is the natural base of logarithms; $\ln u$ denotes the natural logarithm of $u$ where it is assumed that $u > 0$ [in general, to extend formulas to cases where $u < 0$ as well, replace $\ln u$ with $\ln \size u$]; all angles are in radians; all constants of integration are omitted but implied.
- $14.1$: Primitive of $a$
- $14.2$: Primitive of $a \map f x$
- $14.3$: Linear Combination of Primitives: $\ds \int \paren {u \pm v \pm w \pm \cdots} \rd x$
- $14.4$: Integration by Parts: $\ds \int u \rd v$
- $14.5$: Primitive of $\map f {a x}$
- $14.6$: Primitive of $\map F { {\map f x} }$
- $14.7$: Primitive of $u^n$
- $14.8$: Primitive of $\dfrac {\rd u} u$
- $14.9$: Primitive of $e^u$
- $14.10$: Primitive of $a^u$
- $14.11$: Primitive of $\sin u$
- $14.12$: Primitive of $\cos u$
- $14.13$: Primitive of $\tan u$: Secant Form
- Primitive of $\tan u$: Cosine Form
- $14.14$: Primitive of $\cot u$
- $14.15$: Primitive of $\sec u$: Secant plus Tangent Form
- Primitive of $\sec u$: Tangent plus Angle Form
- $14.16$: Primitive of $\csc u$: Cosecant minus Cotangent Form
- Primitive of $\csc u$: Tangent Form
- $14.17$: Primitive of $\sec^2 u$
- $14.18$: Primitive of $\csc^2 u$
- $14.19$: Primitive of $\tan^2 u$
- $14.20$: Primitive of $\cot^2 u$
- $14.21$: Primitive of $\sin^2 u$
- $14.22$: Primitive of $\cos^2 u$
- $14.23$: Primitive of $\sec u \tan u$
- $14.24$: Primitive of $\csc u \cot u$
- $14.25$: Primitive of $\sinh u$
- $14.26$: Primitive of $\cosh u$
- $14.27$: Primitive of $\tanh u$
- $14.28$: Primitive of $\coth u$
- $14.29$: Primitive of $\sech u$
- $14.30$: Primitive of $\csch u$
- $14.31$: Primitive of $\sech^2 u$
- $14.32$: Primitive of $\csch^2 u$
- $14.33$: Primitive of $\tanh^2 u$
- $14.34$: Primitive of $\coth^2 u$
- $14.35$: Primitive of $\sinh^2 u$
- $14.36$: Primitive of $\cosh^2 u$
- $14.37$: Primitive of $\sech u \tanh u$
- $14.38$: Primitive of $\csch u \coth u$
- $14.39$: Primitive of $\dfrac 1 {u^2 + a^2}$
- $14.40$: $u^2 > a^2$:
- Primitive of $\dfrac 1 {u^2 - a^2}$: Logarithm Form
- Primitive of $\dfrac 1 {a^2 - u^2}$: $\coth^{-1}$ Form
- $14.41$: $u^2 < a^2$:
- Primitive of $\dfrac 1 {a^2 - u^2}$: Logarithm Form
- Primitive of $\dfrac 1 {a^2 - u^2}$: $\tanh^{-1}$ Form
- $14.42$: Primitive of $\dfrac 1 {\sqrt {a^2 - u^2} }$
- $14.43$: Primitive of $\dfrac 1 {\sqrt {u^2 + a^2} }$
- $14.44$: Primitive of $\dfrac 1 {\sqrt {u^2 - a^2} }$
- $14.45$: Primitive of $\dfrac 1 {u \sqrt {u^2 - a^2} }$
- $14.46$: Primitive of $\dfrac 1 {u \sqrt {u^2 + a^2} }$
- $14.47$: Primitive of $\dfrac 1 {u \sqrt {a^2 - u^2} }$
- $14.48$: Generalized Integration by Parts: $\ds \int f^{\paren n} g \rd x$
Often in practice an integral can be simplified by using an appropriate transformation or substitution and formula $14.6$. The following list gives some transformations and their effects.
- $14.49$: Primitive of $F \left({a x + b}\right)$
- $14.50$: Primitive of $F \left({ \sqrt{a x + b} }\right)$
- $14.51$: Primitive of $F \left({ \sqrt [n] {a x + b} }\right)$
- $14.52$: Primitive of $F \left({ \sqrt{a^2 - x^2} }\right)$
- $14.53$: Primitive of $F \left({ \sqrt{x^2 + a^2} }\right)$
- $14.54$: Primitive of $F \left({ \sqrt{x^2 - a^2} }\right)$
- $14.55$: Primitive of $F \left({e^{a x}}\right)$
- $14.56$: Primitive of $F \left({\ln x}\right)$
- $14.57$: Primitive of $F \left({\sin^{-1} \dfrac x a}\right)$
- Similar results apply for other inverse trigonometric functions:
- Primitive of $F \left({\cos^{-1} \dfrac x a}\right)$
- Primitive of $F \left({\tan^{-1} \dfrac x a}\right)$
- Primitive of $F \left({\cot^{-1} \dfrac x a}\right)$
- Primitive of $F \left({\sec^{-1} \dfrac x a}\right)$
- Primitive of $F \left({\csc^{-1} \dfrac x a}\right)$
- $14.58$: Primitive of $F \left({\sin x, \cos x}\right)$
Special Integrals
It is assumed in all cases that division by zero is excluded. Also, in the following, $u, v, w$ are functions of $x$; $a, b, p, q, n$ any constants, restricted if indicated; $e = 2.71828 \ldots$ is the natural base of logarithms; $\ln u$ denotes the natural logarithm of $u$ where it is assumed that $u > 0$ [in general, to extend formulas to cases where $u < 0$ as well, replace $\ln u$ with $\ln \size u$]; all angles are in radians; all constants of integration are omitted but implied.
- $14.59$: Primitive of $\dfrac 1 {a x + b}$
- $14.60$: Primitive of $\dfrac x {a x + b}$
- $14.61$: Primitive of $\dfrac {x^2} {a x + b}$
- $14.62$: Primitive of $\dfrac {x^3} {a x + b}$
- $14.63$: Primitive of $\dfrac 1 {x \left({a x + b}\right)}$
- $14.64$: Primitive of $\dfrac 1 {x^2 \left({a x + b}\right)}$
- $14.65$: Primitive of $\dfrac 1 {x^3 \left({a x + b}\right)}$
- $14.66$: Primitive of $\dfrac 1 {\left({a x + b}\right)^2}$
- $14.67$: Primitive of $\dfrac x {\left({a x + b}\right)^2}$
- $14.68$: Primitive of $\dfrac {x^2} {\left({a x + b}\right)^2}$
- $14.69$: Primitive of $\dfrac {x^3} {\left({a x + b}\right)^2}$
- $14.70$: Primitive of $\dfrac 1 {x \left({a x + b}\right)^2}$
- $14.71$: Primitive of $\dfrac 1 {x^2 \left({a x + b}\right)^2}$
- $14.72$: Primitive of $\dfrac 1 {x^3 \left({a x + b}\right)^2}$
- $14.73$: Primitive of $\dfrac 1 {\left({a x + b}\right)^3}$
- $14.74$: Primitive of $\dfrac x {\left({a x + b}\right)^3}$
- $14.75$: Primitive of $\dfrac {x^2} {\left({a x + b}\right)^3}$
- $14.76$: Primitive of $\dfrac {x^3} {\left({a x + b}\right)^3}$
- $14.77$: Primitive of $\dfrac 1 {x \left({a x + b}\right)^3}$
- $14.78$: Primitive of $\dfrac 1 {x^2 \left({a x + b}\right)^3}$
- $14.79$: Primitive of $\dfrac 1 {x^3 \left({a x + b}\right)^3}$
- $14.80$: Primitive of $\left({a x + b}\right)^n$
- $14.81$: Primitive of $x \left({a x + b}\right)^n$
- $14.82$: Primitive of $x^2 \left({a x + b}\right)^n$
- $14.83$: Primitive of $x^m \left({a x + b}\right)^n$:
- Three formulas are given for this case:
- Decrement of Power of $a x + b$
- Decrement of Power of $x$
- Increment of Power of $a x + b$
- The fourth form is not given:
- Increment of Power of $x$
- $14.84$: Primitive of $\dfrac 1 {\sqrt{a x + b}}$
- $14.85$: Primitive of $\dfrac x {\sqrt{a x + b}}$
- $14.86$: Primitive of $\dfrac {x^2} {\sqrt{a x + b}}$
- $14.87$: Primitive of $\dfrac 1 {x \sqrt{a x + b}}$
- $14.88$: Primitive of $\dfrac 1 {x^2 \sqrt{a x + b}}$
- $14.89$: Primitive of $\sqrt{a x + b}$
- $14.90$: Primitive of $x \sqrt{a x + b}$
- $14.91$: Primitive of $x^2 \sqrt{a x + b}$
- $14.92$: Primitive of $\dfrac {\sqrt{a x + b}} x$
- $14.93$: Primitive of $\dfrac {\sqrt{a x + b}} {x^2}$
- $14.94$: Primitive of $\dfrac {x^m} {\sqrt{a x + b}}$
- $14.95$: Primitive of $\dfrac 1 {x^m \sqrt{a x + b}}$
- $14.96$: Primitive of $x^m \sqrt{a x + b}$
- $14.97-98$: Primitive of $\dfrac {\sqrt{a x + b}} {x^m}$
- Two formulations are given:
- $14.97$: Formulation 1
- $14.98$: Formulation 2
- $14.99$: Primitive of $\left({a x + b}\right)^{m / 2}$
- $14.100$: Primitive of $x \left({a x + b}\right)^{m / 2}$
- $14.101$: Primitive of $x^2 \left({a x + b}\right)^{m / 2}$
- $14.102$: Primitive of $\dfrac {\left({a x + b}\right)^{m / 2}} x$
- $14.103$: Primitive of $\dfrac {\left({a x + b}\right)^{m / 2} } {x^2}$
- $14.104$: Primitive of $\dfrac 1 {x \left({a x + b}\right)^{m / 2} }$
- $14.105$: Primitive of $\dfrac 1 {\left({a x + b}\right) \left({p x + q}\right)}$
- $14.106$: Primitive of $\dfrac x {\left({a x + b}\right) \left({p x + q}\right)}$
- $14.107$: Primitive of $\dfrac 1 {\left({a x + b}\right)^2 \left({p x + q}\right)}$
- $14.108$: Primitive of $\dfrac x {\left({a x + b}\right)^2 \left({p x + q}\right)}$
- $14.109$: Primitive of $\dfrac {x^2} {\left({a x + b}\right)^2 \left({p x + q}\right)}$
- $14.110$: Primitive of $\dfrac 1 {\left({a x + b}\right)^m \left({p x + q}\right)^n}$
- $14.111$: Primitive of $\dfrac {a x + b} {p x + q}$
- $14.112$: Primitive of $\dfrac {\left({a x + b}\right)^m} {\left({p x + q}\right)^n}$
- $14.113$: Primitive of $\dfrac {p x + q} {\sqrt{a x + b}}$
- $14.114$: Primitive of $\dfrac 1 {\left({p x + q}\right) \sqrt{a x + b}}$
- $14.115$: Primitive of $\dfrac {\sqrt{a x + b}} {p x + q}$
- $14.116$: Primitive of $\left({p x + q}\right)^n \sqrt{a x + b}$
- $14.117$: Primitive of $\dfrac 1 {\left({p x + q}\right)^n \sqrt{a x + b}}$
- $14.118$: Primitive of $\dfrac {\left({p x + q}\right)^n} {\sqrt{a x + b}}$
- $14.119$: Primitive of $\dfrac {\sqrt{a x + b}} {\left({p x + q}\right)^n}$
- $14.120$: Primitive of $\dfrac 1 {\sqrt{\left({a x + b}\right) \left({p x + q}\right)}}$
- $14.121$: Primitive of $\dfrac x {\sqrt{\left({a x + b}\right) \left({p x + q}\right)}}$
- $14.122$: Primitive of $\sqrt{\left({a x + b}\right) \left({p x + q}\right)}$
- $14.123$: Primitive of $\sqrt{\dfrac {\left({a x + b}\right)} {\left({p x + q}\right)} }$
- $14.124$: Primitive of $\dfrac 1 {\left({p x + q}\right) \sqrt{\left({a x + b}\right) \left({p x + q}\right)}}$
- $14.125$: Primitive of $\dfrac 1 {x^2 + a^2}$
- $14.126$: Primitive of $\dfrac x {x^2 + a^2}$
- $14.127$: Primitive of $\dfrac {x^2} {x^2 + a^2}$
- $14.128$: Primitive of $\dfrac {x^3} {x^2 + a^2}$
- $14.129$: Primitive of $\dfrac 1 {x \paren {x^2 + a^2} }$
- $14.130$: Primitive of $\dfrac 1 {x^2 \paren {x^2 + a^2} }$
- $14.131$: Primitive of $\dfrac 1 {x^3 \paren {x^2 + a^2}}$
- $14.132$: Primitive of $\dfrac 1 {\paren {x^2 + a^2}^2}$
- $14.133$: Primitive of $\dfrac x {\paren {x^2 + a^2}^2}$
- $14.134$: Primitive of $\dfrac {x^2} {\paren {x^2 + a^2}^2}$
- $14.135$: Primitive of $\dfrac {x^3} {\paren {x^2 + a^2}^2}$
- $14.136$: Primitive of $\dfrac 1 {x \paren {x^2 + a^2}^2}$
- $14.137$: Primitive of $\dfrac 1 {x^2 \paren {x^2 + a^2}^2}$
- $14.138$: Primitive of $\dfrac 1 {x^3 \paren {x^2 + a^2}^2}$
- $14.140$: Primitive of $\dfrac 1 {\paren {x^2 + a^2}^n}$
- $14.141$: Primitive of $\dfrac 1 {x \paren {x^2 + a^2}^n}$
- $14.142$: Primitive of $\dfrac {x^m} {\paren {x^2 + a^2}^n}$
- $14.143$: Primitive of $\dfrac 1 {x^m \paren {x^2 + a^2}^n}$
- $14.144$
- Primitive of $\dfrac 1 {x^2 - a^2}$: Logarithm Form
- Primitive of $\dfrac 1 {x^2 - a^2}$: $\coth^{-1}$ Form
- $14.145$: Primitive of $\dfrac x {x^2 - a^2}$
- $14.146$: Primitive of $\dfrac {x^2} {x^2 - a^2}$
- $14.147$: Primitive of $\dfrac {x^3} {x^2 - a^2}$
- $14.148$: Primitive of $\dfrac 1 {x \paren {x^2 - a^2} }$
- $14.149$: Primitive of $\dfrac 1 {x^2 \paren {x^2 - a^2} }$
- $14.150$: Primitive of $\dfrac 1 {x^3 \paren {x^2 - a^2} }$
- $14.151$: Primitive of $\dfrac 1 {\paren {x^2 - a^2}^2}$
- $14.152$: Primitive of $\dfrac x {\paren {x^2 - a^2}^2}$
- $14.153$: Primitive of $\dfrac {x^2} {\paren {x^2 - a^2}^2}$
- $14.154$: Primitive of $\dfrac {x^3} {\paren {x^2 - a^2}^2}$
- $14.155$: Primitive of $\dfrac 1 {x \paren {x^2 - a^2}^2}$
- $14.156$: Primitive of $\dfrac 1 {x^2 \paren {x^2 - a^2}^2}$
- $14.157$: Primitive of $\dfrac 1 {x^3 \paren {x^2 - a^2}^2}$
- $14.158$: Primitive of $\dfrac 1 {\paren {x^2 - a^2}^n}$
- $14.159$: Primitive of $\dfrac x {\paren {x^2 - a^2}^n}$
- $14.160$: Primitive of $\dfrac 1 {x \paren {x^2 - a^2}^n}$
- $14.161$: Primitive of $\dfrac {x^m} {\paren {x^2 - a^2}^n}$
- $14.162$: Primitive of $\dfrac 1 {x^m \paren {x^2 - a^2}^n}$
- $14.163$:
- Primitive of $\dfrac 1 {a^2 - x^2}$: Logarithm Form
- Primitive of $\dfrac 1 {a^2 - x^2}$: $\tanh^{-1}$ form
- $14.164$: Primitive of $\dfrac x {a^2 - x^2}$
- $14.165$: Primitive of $\dfrac {x^2} {a^2 - x^2}$
- $14.166$: Primitive of $\dfrac {x^3} {a^2 - x^2}$
- $14.167$: Primitive of $\dfrac 1 {x \left({a^2 - x^2}\right)}$
- $14.168$: Primitive of $\dfrac 1 {x^2 \left({a^2 - x^2}\right)}$
- $14.169$: Primitive of $\dfrac 1 {x^3 \left({a^2 - x^2}\right)}$
- $14.170$: Primitive of $\dfrac 1 {\left({a^2 - x^2}\right)^2}$
- $14.171$: Primitive of $\dfrac x {\left({a^2 - x^2}\right)^2}$
- $14.172$: Primitive of $\dfrac {x^2} {\left({a^2 - x^2}\right)^2}$
- $14.173$: Primitive of $\dfrac {x^3} {\left({a^2 - x^2}\right)^2}$
- $14.174$: Primitive of $\dfrac 1 {x \left({a^2 - x^2}\right)^2}$
- $14.175$: Primitive of $\dfrac 1 {x^2 \left({a^2 - x^2}\right)^2}$
- $14.176$: Primitive of $\dfrac 1 {x^3 \left({a^2 - x^2}\right)^2}$
- $14.177$: Primitive of $\dfrac 1 {\left({a^2 - x^2}\right)^n}$
- $14.178$: Primitive of $\dfrac x {\left({a^2 - x^2}\right)^n}$
- $14.179$: Primitive of $\dfrac 1 {x \left({a^2 - x^2}\right)^n}$
- $14.180$: Primitive of $\dfrac {x^m} {\left({a^2 - x^2}\right)^n}$
- $14.181$: Primitive of $\dfrac 1 {x^m \left({a^2 - x^2}\right)^n}$
- $14.182$: Primitive of $\dfrac 1 {\sqrt{x^2 + a^2}}$
- $14.183$: Primitive of $\dfrac x {\sqrt{x^2 + a^2}}$
- $14.184$: Primitive of $\dfrac {x^2} {\sqrt{x^2 + a^2}}$
- $14.185$: Primitive of $\dfrac {x^3} {\sqrt{x^2 + a^2}}$
- $14.186$: Primitive of $\dfrac 1 {x \left({\sqrt{x^2 + a^2}}\right)}$
- $14.187$: Primitive of $\dfrac 1 {x^2 \left({\sqrt{x^2 + a^2}}\right)}$
- $14.188$: Primitive of $\dfrac 1 {x^3 \left({\sqrt{x^2 + a^2}}\right)}$
- $14.189$: Primitive of $\sqrt{x^2 + a^2}$
- $14.190$: Primitive of $x \sqrt{x^2 + a^2}$
- $14.191$: Primitive of $x^2 \sqrt{x^2 + a^2}$
- $14.192$: Primitive of $x^3 \sqrt{x^2 + a^2}$
- $14.193$: Primitive of $\dfrac {\sqrt{x^2 + a^2}} x$
- $14.194$: Primitive of $\dfrac {\sqrt{x^2 + a^2}} {x^2}$
- $14.195$: Primitive of $\dfrac {\sqrt{x^2 + a^2}} {x^3}$
- $14.196$: Primitive of $\dfrac 1 {\left({x^2 + a^2}\right)^{3/2}}$
- $14.197$: Primitive of $\dfrac x {\left({x^2 + a^2}\right)^{3/2}}$
- $14.198$: Primitive of $\dfrac {x^2} {\left({x^2 + a^2}\right)^{3/2}}$
- $14.199$: Primitive of $\dfrac {x^3} {\left({x^2 + a^2}\right)^{3/2}}$
- $14.200$: Primitive of $\dfrac 1 {x \left({x^2 + a^2}\right)^{3/2}}$
- $14.201$: Primitive of $\dfrac 1 {x^2 \left({x^2 + a^2}\right)^{3/2}}$
- $14.202$: Primitive of $\dfrac 1 {x^3 \left({x^2 + a^2}\right)^{3/2}}$
- $14.203$: Primitive of $\left({x^2 + a^2}\right)^{3/2}$
- $14.204$: Primitive of $x \left({x^2 + a^2}\right)^{3/2}$
- $14.205$: Primitive of $x^2 \left({x^2 + a^2}\right)^{3/2}$
- $14.206$: Primitive of $x^3 \left({x^2 + a^2}\right)^{3/2}$
- $14.207$: Primitive of $\dfrac {\left({x^2 + a^2}\right)^{3/2}} x$
- $14.208$: Primitive of $\dfrac {\left({x^2 + a^2}\right)^{3/2}} {x^2}$
- $14.209$: Primitive of $\dfrac {\left({x^2 + a^2}\right)^{3/2}} {x^3}$
- $14.210.1$: Primitive of $\dfrac 1 {\sqrt{x^2 - a^2}}$
- $14.210.2$: Primitive of $\dfrac x {\sqrt{x^2 - a^2}}$
- $14.211$: Primitive of $\dfrac {x^2} {\sqrt{x^2 - a^2}}$
- $14.212$: Primitive of $\dfrac {x^3} {\sqrt{x^2 - a^2}}$
- $14.213$: Primitive of $\dfrac 1 {x \left({\sqrt{x^2 - a^2}}\right)}$
- $14.214$: Primitive of $\dfrac 1 {x^2 \left({\sqrt{x^2 - a^2}}\right)}$
- $14.215$: Primitive of $\dfrac 1 {x^3 \left({\sqrt{x^2 - a^2}}\right)}$
- $14.216$: Primitive of $\sqrt{x^2 - a^2}$
- $14.217$: Primitive of $x \sqrt{x^2 - a^2}$
- $14.218$: Primitive of $x^2 \sqrt{x^2 - a^2}$
- $14.219$: Primitive of $x^3 \sqrt{x^2 - a^2}$
- $14.220$: Primitive of $\dfrac {\sqrt{x^2 - a^2}} x$
- $14.221$: Primitive of $\dfrac {\sqrt{x^2 - a^2}} {x^2}$
- $14.222$: Primitive of $\dfrac {\sqrt{x^2 - a^2}} {x^3}$
- $14.223$: Primitive of $\dfrac 1 {\left({x^2 - a^2}\right)^{3/2}}$
- $14.224$: Primitive of $\dfrac x {\left({x^2 - a^2}\right)^{3/2}}$
- $14.225$: Primitive of $\dfrac {x^2} {\left({x^2 - a^2}\right)^{3/2}}$
- $14.226$: Primitive of $\dfrac {x^3} {\left({x^2 - a^2}\right)^{3/2}}$
- $14.227$: Primitive of $\dfrac 1 {x \left({x^2 - a^2}\right)^{3/2}}$
- $14.228$: Primitive of $\dfrac 1 {x^2 \left({x^2 - a^2}\right)^{3/2}}$
- $14.229$: Primitive of $\dfrac 1 {x^3 \left({x^2 - a^2}\right)^{3/2}}$
- $14.230$: Primitive of $\left({x^2 - a^2}\right)^{3/2}$
- $14.231$: Primitive of $x \left({x^2 - a^2}\right)^{3/2}$
- $14.232$: Primitive of $x^2 \left({x^2 - a^2}\right)^{3/2}$
- $14.233$: Primitive of $x^3 \left({x^2 - a^2}\right)^{3/2}$
- $14.234$: Primitive of $\dfrac {\left({x^2 - a^2}\right)^{3/2}} x$
- $14.235$: Primitive of $\dfrac {\left({x^2 - a^2}\right)^{3/2}} {x^2}$
- $14.236$: Primitive of $\dfrac {\left({x^2 - a^2}\right)^{3/2}} {x^3}$
- $14.237$: Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$
- $14.238$: Primitive of $\dfrac x {\sqrt {a^2 - x^2} }$
- $14.239$: Primitive of $\dfrac {x^2} {\sqrt {a^2 - x^2} }$
- $14.240$: Primitive of $\dfrac {x^3} {\sqrt {a^2 - x^2} }$
- $14.241$: Primitive of $\dfrac 1 {x \paren {\sqrt {a^2 - x^2} } }$
- $14.242$: Primitive of $\dfrac 1 {x^2 \paren {\sqrt {a^2 - x^2} } }$
- $14.243$: Primitive of $\dfrac 1 {x^3 \paren {\sqrt {a^2 - x^2} } }$
- $14.244$: Primitive of $\sqrt {a^2 - x^2}$
- $14.245$: Primitive of $x \sqrt {a^2 - x^2}$
- $14.246$: Primitive of $x^2 \sqrt {a^2 - x^2}$
- $14.247$: Primitive of $x^3 \sqrt {a^2 - x^2}$
- $14.248$: Primitive of $\dfrac {\sqrt {a^2 - x^2} } x$
- $14.249$: Primitive of $\dfrac {\sqrt {a^2 - x^2} } {x^2}$
- $14.250$: Primitive of $\dfrac {\sqrt {a^2 - x^2} } {x^3}$
- $14.251$: Primitive of $\dfrac 1 {\paren {a^2 - x^2}^{3/2} }$
- $14.252$: Primitive of $\dfrac x {\paren {a^2 - x^2}^{3/2} }$
- $14.253$: Primitive of $\dfrac {x^2} {\paren {a^2 - x^2}^{3/2} }$
- $14.254$: Primitive of $\dfrac {x^3} {\paren {a^2 - x^2}^{3/2} }$
- $14.255$: Primitive of $\dfrac 1 {x \paren {a^2 - x^2}^{3/2} }$
- $14.256$: Primitive of $\dfrac 1 {x^2 \paren {a^2 - x^2}^{3/2} }$
- $14.257$: Primitive of $\dfrac 1 {x^3 \paren {a^2 - x^2}^{3/2} }$
- $14.258$: Primitive of $\paren {a^2 - x^2}^{3/2}$
- $14.259$: Primitive of $x \paren {a^2 - x^2}^{3/2}$
- $14.260$: Primitive of $x^2 \paren {a^2 - x^2}^{3/2}$
- $14.261$: Primitive of $x^3 \paren {a^2 - x^2}^{3/2}$
- $14.262$: Primitive of $\dfrac {\paren {a^2 - x^2}^{3/2} } x$
- $14.263$: Primitive of $\dfrac {\paren {a^2 - x^2}^{3/2} } {x^2}$
- $14.264$: Primitive of $\dfrac {\paren {a^2 - x^2}^{3/2} } {x^3}$
- $14.265$: Primitive of $\dfrac 1 {a x^2 + b x + c}$
If $b^2 = 4 a c$, $a x^2 + b x + c = a \left({x + b / 2 a}\right)^2$ and the results from Integrals Involving $a x + b$ can be used. If $b = 0$ use results from Integrals Involving $x^2 + a^2$. If $a$ or $c = 0$ use results from Integrals Involving $a x + b$.
- $14.266$: Primitive of $\dfrac x {a x^2 + b x + c}$
- $14.267$: Primitive of $\dfrac {x^2} {a x^2 + b x + c}$
- $14.268$: Primitive of $\dfrac {x^m} {a x^2 + b x + c}$
- $14.269$: Primitive of $\dfrac 1 {x \left({a x^2 + b x + c}\right)}$
- $14.270$: Primitive of $\dfrac 1 {x^2 \left({a x^2 + b x + c}\right)}$
- $14.271$: Primitive of $\dfrac 1 {x^n \left({a x^2 + b x + c}\right)}$
- $14.272$: Primitive of $\dfrac 1 {\left({a x^2 + b x + c}\right)^2}$
- $14.273$: Primitive of $\dfrac x {\left({a x^2 + b x + c}\right)^2}$
- $14.274$: Primitive of $\dfrac {x^2} {\left({a x^2 + b x + c}\right)^2}$
- $14.275$: Primitive of $\dfrac {x^m} {\left({a x^2 + b x + c}\right)^n}$
- $14.276$: Primitive of $\dfrac {x^{2 n - 1}} {\left({a x^2 + b x + c}\right)^n}$
- $14.277$: Primitive of $\dfrac 1 {x \left({a x^2 + b x + c}\right)^2}$
- $14.278$: Primitive of $\dfrac 1 {x^2 \left({a x^2 + b x + c}\right)^2}$
- $14.279$: Primitive of $\dfrac 1 {x^m \left({a x^2 + b x + c}\right)^n}$
In the following results if $b^2 = 4 a c$, $\sqrt{a x^2 + b x + c} = \sqrt a \left({x + b / 2 a}\right)$ and the results from Integrals Involving $a x + b$ can be used. If $b = 0$ use results from Integrals Involving $\sqrt {x^2 + a^2}$, Integrals Involving $\sqrt {x^2 - a^2}$, $x^2 > a^2$ and Integrals Involving $\sqrt {a^2 - x^2}$, $a^2 > x^2$. If $a$ or $c = 0$ use results from Integrals Involving $\sqrt{a x + b}$.
- $14.280$: Primitive of $\dfrac 1 {\sqrt{a x^2 + b x + c}}$
- $14.281$: Primitive of $\dfrac x {\sqrt{a x^2 + b x + c}}$
- $14.282$: Primitive of $\dfrac {x^2} {\sqrt{a x^2 + b x + c}}$
- $14.283$: Primitive of $\dfrac 1 {x \left({\sqrt{a x^2 + b x + c}}\right)}$
- $14.284$: Primitive of $\dfrac 1 {x^2 \left({\sqrt{a x^2 + b x + c}}\right)}$
- $14.285$: Primitive of $\sqrt{a x^2 + b x + c}$
- $14.286$: Primitive of $x \sqrt{a x^2 + b x + c}$
- $14.287$: Primitive of $x^2 \sqrt{a x^2 + b x + c}$
- $14.288$: Primitive of $\dfrac {\sqrt{a x^2 + b x + c}} x$
- $14.289$: Primitive of $\dfrac {\sqrt{a x^2 + b x + c}} {x^2}$
- $14.290$: Primitive of $\dfrac 1 {\left({a x^2 + b x + c}\right)^{3/2}}$
- $14.291$: Primitive of $\dfrac x {\left({a x^2 + b x + c}\right)^{3/2}}$
- $14.292$: Primitive of $\dfrac {x^2} {\left({a x^2 + b x + c}\right)^{3/2}}$
- $14.293$: Primitive of $\dfrac 1 {x \left({a x^2 + b x + c}\right)^{3/2}}$
- $14.294$: Primitive of $\dfrac 1 {x^2 \left({a x^2 + b x + c}\right)^{3/2}}$
- $14.295$: Primitive of $\left({a x^2 + b x + c}\right)^{n + 1 / 2}$
- $14.296$: Primitive of $x \left({a x^2 + b x + c}\right)^{n + 1 / 2}$
- $14.297$: Primitive of $\dfrac 1 {\left({a x^2 + b x + c}\right)^{n + 1 / 2}}$
- $14.298$: Primitive of $\dfrac 1 {x \left({a x^2 + b x + c}\right)^{n + 1 / 2}}$
Note that for formulas involving $x^3 - a^3$ replace $a$ by $-a$.
- $14.299$: Primitive of $\dfrac 1 {x^3 + a^3}$
- $14.300$: Primitive of $\dfrac x {x^3 + a^3}$
- $14.301$: Primitive of $\dfrac {x^2} {x^3 + a^3}$
- $14.302$: Primitive of $\dfrac 1 {x \left({x^3 + a^3}\right)}$
- $14.303$: Primitive of $\dfrac 1 {x^2 \left({x^3 + a^3}\right)}$
- $14.304$: Primitive of $\dfrac 1 {\left({x^3 + a^3}\right)^2}$
- $14.305$: Primitive of $\dfrac x {\left({x^3 + a^3}\right)^2}$
- $14.306$: Primitive of $\dfrac {x^2} {\left({x^3 + a^3}\right)^2}$
- $14.307$: Primitive of $\dfrac 1 {x \left({x^3 + a^3}\right)^2}$
- $14.308$: Primitive of $\dfrac 1 {x^2 \left({x^3 + a^3}\right)^2}$
- $14.309$: Primitive of $\dfrac {x^m} {x^3 + a^3}$
- $14.310$: Primitive of $\dfrac 1 {x^n \left({x^3 + a^3}\right)}$
- $14.311$: Primitive of $\dfrac 1 {x^4 + a^4}$
- $14.312$: Primitive of $\dfrac x {x^4 + a^4}$
- $14.313$: Primitive of $\dfrac {x^2} {x^4 + a^4}$
- $14.314$: Primitive of $\dfrac {x^3} {x^4 + a^4}$
- $14.315$: Primitive of $\dfrac 1 {x \left({x^4 + a^4}\right)}$
- $14.316$: Primitive of $\dfrac 1 {x^2 \left({x^4 + a^4}\right)}$
- $14.317$: Primitive of $\dfrac 1 {x^3 \left({x^4 + a^4}\right)}$
- $14.318$: Primitive of $\dfrac 1 {x^4 - a^4}$
- $14.319$: Primitive of $\dfrac x {x^4 - a^4}$
- $14.320$: Primitive of $\dfrac {x^2} {x^4 - a^4}$
- $14.321$: Primitive of $\dfrac {x^3} {x^4 - a^4}$
- $14.322$: Primitive of $\dfrac 1 {x \left({x^4 - a^4}\right)}$
- $14.323$: Primitive of $\dfrac 1 {x^2 \left({x^4 - a^4}\right)}$
- $14.324$: Primitive of $\dfrac 1 {x^3 \left({x^4 - a^4}\right)}$
- $14.325$: Primitive of $\dfrac 1 {x \left({x^n + a^n}\right)}$
- $14.326$: Primitive of $\dfrac {x^{n - 1}} {x \left({x^n + a^n}\right)}$
- $14.327$: Primitive of $\dfrac {x^m} {\left({x^n + a^n}\right)^r}$
- $14.328$: Primitive of $\dfrac 1 {x^m \left({x^n + a^n}\right)^r}$
- $14.329$: Primitive of $\dfrac 1 {x \sqrt{x^n + a^n}}$
- $14.330$: Primitive of $\dfrac 1 {x \left({x^n - a^n}\right)}$
- $14.331$: Primitive of $\dfrac {x^{n - 1}} {x \left({x^n - a^n}\right)}$
- $14.332$: Primitive of $\dfrac {x^m} {\left({x^n - a^n}\right)^r}$
- $14.333$: Primitive of $\dfrac 1 {x^m \left({x^n - a^n}\right)^r}$
- $14.334$: Primitive of $\dfrac 1 {x \sqrt{x^n - a^n}}$
- $14.335$: Primitive of $\dfrac {x^{p - 1}} {x^{2 m} + a^{2 m}}$
- $14.336$: Primitive of $\dfrac {x^{p - 1}} {x^{2 m} - a^{2 m}}$
- $14.337$: Primitive of $\dfrac {x^{p - 1}} {x^{2 m + 1} + a^{2 m + 1}}$
- $14.338$: Primitive of $\dfrac {x^{p - 1}} {x^{2 m + 1} - a^{2 m + 1}}$
- $14.339$: Primitive of $\sin a x$
- $14.340$: Primitive of $x \sin a x$
- $14.341$: Primitive of $x^2 \sin a x$
- $14.342$: Primitive of $x^3 \sin a x$
- $14.343$: Primitive of $\dfrac {\sin a x} x$
- $14.344$: Primitive of $\dfrac {\sin a x} {x^2}$
- $14.345$: Primitive of $\dfrac 1 {\sin a x}$
- $14.346$: Primitive of $\dfrac x {\sin a x}$
- $14.347$: Primitive of $\sin^2 a x$
- $14.348$: Primitive of $x \sin^2 a x$
- $14.349$: Primitive of $\sin^3 a x$
- $14.350$: Primitive of $\sin^4 a x$
- $14.351$: Primitive of $\dfrac 1 {\sin^2 a x}$
- $14.352$: Primitive of $\dfrac 1 {\sin^3 a x}$
- $14.353$: Primitive of $\sin p x \sin q x$
- [If $p = \pm q$, see $14.347$: Primitive of $\sin^2 a x$.]
- $14.354$: Primitive of $\dfrac 1 {1 - \sin a x}$
- $14.355$: Primitive of $\dfrac x {1 - \sin a x}$
- $14.356$: Primitive of $\dfrac 1 {1 + \sin a x}$
- $14.357$: Primitive of $\dfrac x {1 + \sin a x}$
- $14.358$: Primitive of $\dfrac 1 {\left({1 - \sin a x}\right)^2}$
- $14.359$: Primitive of $\dfrac 1 {\left({1 + \sin a x}\right)^2}$
- $14.360$: Primitive of $\dfrac 1 {p + q \sin a x}$
- [If $p = \pm q$, see $14.354$: Primitive of $\dfrac 1 {1 - \sin a x}$ and $14.356$: Primitive of $\dfrac 1 {1 + \sin a x}$.]
- $14.361$: Primitive of $\dfrac 1 {\left({p + q \sin a x}\right)^2}$
- [If $p = \pm q$, see $14.358$: Primitive of $\dfrac 1 {\left({1 - \sin a x}\right)^2}$ and $14.359$: Primitive of $\dfrac 1 {\left({1 + \sin a x}\right)^2}$.]
- $14.362$: Primitive of $\dfrac 1 {p^2 + q^2 \sin^2 a x}$
- $14.363$: Primitive of $\dfrac 1 {p^2 - q^2 \sin^2 a x}$
- $14.364$: Primitive of $x^m \sin a x$
- $14.365$: Primitive of $\dfrac {\sin a x} {x^n}$
- $14.366$: Primitive of $\sin^n a x$
- $14.367$: Primitive of $\dfrac 1 {\sin^n a x}$
- $14.368$: Primitive of $\dfrac x {\sin^n a x}$
- $14.369$: Primitive of $\cos a x$
- $14.370$: Primitive of $x \cos a x$
- $14.371$: Primitive of $x^2 \cos a x$
- $14.372$: Primitive of $x^3 \cos a x$
- $14.373$: Primitive of $\dfrac {\cos a x} x$
- $14.374$: Primitive of $\dfrac {\cos a x} {x^2}$
- $14.375$: Primitive of $\dfrac 1 {\cos a x}$
- $14.376$: Primitive of $\dfrac x {\cos a x}$
- $14.377$: Primitive of $\cos^2 a x$
- $14.378$: Primitive of $x \cos^2 a x$
- $14.379$: Primitive of $\cos^3 a x$
- $14.380$: Primitive of $\cos^4 a x$
- $14.381$: Primitive of $\dfrac 1 {\cos^2 a x}$
- $14.382$: Primitive of $\dfrac 1 {\cos^3 a x}$
- $14.383$: Primitive of $\cos a x \cos p x$
- [If $a = \pm p$, see $14.377$: Primitive of $\cos^2 a x$.]
- $14.384$: Primitive of $\dfrac 1 {1 - \cos a x}$
- $14.385$: Primitive of $\dfrac x {1 - \cos a x}$
- $14.386$: Primitive of $\dfrac 1 {1 + \cos a x}$
- $14.387$: Primitive of $\dfrac x {1 + \cos a x}$
- $14.388$: Primitive of $\dfrac 1 {\left({1 - \cos a x}\right)^2}$
- $14.389$: Primitive of $\dfrac 1 {\left({1 + \cos a x}\right)^2}$
- $14.390$: Primitive of $\dfrac 1 {p + q \cos a x}$
- [If $p = \pm q$, see $14.384$: Primitive of $\dfrac 1 {1 - \cos a x}$ and $14.386$: Primitive of $\dfrac 1 {1 + \cos a x}$.]
- $14.391$: Primitive of $\dfrac 1 {\left({p + q \cos a x}\right)^2}$
- [If $p = \pm q$, see $14.388$: Primitive of $\dfrac 1 {\left({1 - \cos a x}\right)^2}$ and $14.389$: Primitive of $\dfrac 1 {\left({1 + \cos a x}\right)^2}$.]
- $14.392$: Primitive of $\dfrac 1 {p^2 + q^2 \cos^2 a x}$
- $14.393$: Primitive of $\dfrac 1 {p^2 - q^2 \cos^2 a x}$
- $14.394$: Primitive of $x^m \cos a x$
- $14.395$: Primitive of $\dfrac {\cos a x} {x^n}$
- $14.396$: Primitive of $\cos^n a x$
- $14.397$: Primitive of $\dfrac 1 {\cos^n a x}$
- $14.398$: Primitive of $\dfrac x {\cos^n a x}$
- $14.399$: Primitive of $\sin a x \cos a x$
- $14.400$: Primitive of $\sin p x \cos q x$
- $14.401$: Primitive of $\sin^n a x \cos a x$
- $14.402$: Primitive of $\cos^n a x \sin a x$
- $14.403$: Primitive of $\sin^2 a x \cos^2 a x$
- $14.404$: Primitive of $\dfrac 1 {\sin a x \cos a x}$
- $14.405$: Primitive of $\dfrac 1 {\sin^2 a x \cos a x}$
- $14.406$: Primitive of $\dfrac 1 {\sin a x \cos^2 a x}$
- $14.407$: Primitive of $\dfrac 1 {\sin^2 a x \cos^2 a x}$
- $14.408$: Primitive of $\dfrac {\sin^2 a x} {\cos a x}$
- $14.409$: Primitive of $\dfrac {\cos^2 a x} {\sin a x}$
- $14.410$: Primitive of $\dfrac 1 {\cos a x \left({1 \pm \sin a x}\right)}$
- $14.410.1$: Primitive of $\dfrac 1 {\cos a x \left({1 + \sin a x}\right)}$
- $14.410.2$: Primitive of $\dfrac 1 {\cos a x \left({1 - \sin a x}\right)}$
- $14.411$: Primitive of $\dfrac 1 {\sin a x \left({1 \pm \cos a x}\right)}$
- $14.411.1$: Primitive of $\dfrac 1 {\sin a x \left({1 + \cos a x}\right)}$
- $14.411.2$: Primitive of $\dfrac 1 {\sin a x \left({1 - \cos a x}\right)}$
- $14.412$: Primitive of $\dfrac 1 {\sin a x \pm \cos a x}$
- $14.412.1$: Primitive of $\dfrac 1 {\sin a x + \cos a x}$
- $14.412.2$: Primitive of $\dfrac 1 {\sin a x \left({1 - \cos a x}\right)}$
- $14.413$: Primitive of $\dfrac {\sin a x} {\sin a x \pm \cos a x}$
- $14.413.1$: Primitive of $\dfrac {\sin a x} {\sin a x + \cos a x}$
- $14.413.2$: Primitive of $\dfrac {\sin a x} {\sin a x \left({1 - \cos a x}\right)}$
- $14.414$: Primitive of $\dfrac {\cos a x} {\sin a x \pm \cos a x}$
- $14.414.1$: Primitive of $\dfrac {\cos a x} {\sin a x + \cos a x}$
- $14.414.2$: Primitive of $\dfrac {\cos a x} {\sin a x \left({1 - \cos a x}\right)}$
- $14.415$: Primitive of $\dfrac {\sin a x} {p + q \cos a x}$
- $14.416$: Primitive of $\dfrac {\cos a x} {p + q \sin a x}$
- $14.417$: Primitive of $\dfrac {\sin a x} {\left({p + q \cos a x}\right)^n}$
- $14.418$: Primitive of $\dfrac {\cos a x} {\left({p + q \cos a x}\right)^n}$
- $14.419$: Primitive of $\dfrac 1 {p \sin a x + q \cos a x}$
- $14.420$: Primitive of $\dfrac 1 {p \sin a x + q \cos a x + r}$
- [If $r = q$, see $14.421$: Primitive of $\dfrac 1 {p \sin a x + q \left({1 + \cos a x}\right)}$.
- If $r^2 = p^2 + q^2$, see $14.422$: Primitive of $\dfrac 1 {p \sin a x + q \cos a x \pm \sqrt{p^2 + q^2}}$.]
- $14.421$: Primitive of $\dfrac 1 {p \sin a x + q \left({1 + \cos a x}\right)}$
- $14.422$: Primitive of $\dfrac 1 {p \sin a x + q \cos a x \pm \sqrt{p^2 + q^2}}$
- $14.422.1$: Primitive of $\dfrac 1 {p \sin a x + q \cos a x + \sqrt{p^2 + q^2}}$
- $14.422.2$: Primitive of $\dfrac 1 {p \sin a x + q \cos a x - \sqrt{p^2 + q^2}}$
- $14.423$: Primitive of $\dfrac 1 {p^2 \sin^2 a x + q^2 \cos^2 a x}$
- $14.424$: Primitive of $\dfrac 1 {p^2 \sin^2 a x - q^2 \cos^2 a x}$
- $14.425$: Primitive of $\sin^m a x \cos^n a x$
- $14.425.1$: Reduction of Power of Sine
- $14.425.2$: Reduction of Power of Cosine
- $14.426$: Primitive of $\dfrac {\sin^m a x} {\cos^n a x}$
- $14.426.1$: Reduction of Both Powers
- $14.426.2$: Reduction of Power of Sine
- $14.426.3$: Reduction of Power of Cosine
- $14.427$: Primitive of $\dfrac {\cos^m a x} {\sin^n a x}$
- $14.427.1$: Reduction of Both Powers
- $14.427.2$: Reduction of Power of Sine
- $14.427.3$: Reduction of Power of Cosine
- $14.428$: Primitive of $\dfrac 1 {\sin^m a x \cos^n a x}$
- $14.428.1$: Reduction of Power of Cosine
- $14.428.2$: Reduction of Power of Sine
- $14.429$: Primitive of $\tan a x$
- $14.430$: Primitive of $\tan^2 a x$
- $14.431$: Primitive of $\tan^3 a x$
- $14.432$: Primitive of $\tan^n a x \sec^2 a x$
- $14.433$: Primitive of $\dfrac {\sec^2 a x} {\tan a x}$
- $14.434$: Primitive of $\dfrac 1 {\tan a x}$
- $14.435$: Primitive of $x \tan a x$
- $14.436$: Primitive of $\dfrac {\tan a x} x$
- $14.437$: Primitive of $x \tan^2 a x$
- $14.438$: Primitive of $\dfrac 1 {p + q \tan a x}$
- $14.439$: Primitive of $\tan^n a x$
- $14.440$: Primitive of $\cot a x$
- $14.441$: Primitive of $\cot^2 a x$
- $14.442$: Primitive of $\cot^3 a x$
- $14.443$: Primitive of $\cot^n a x \csc^2 a x$
- $14.444$: Primitive of $\dfrac {\csc^2 a x} {\cot a x}$
- $14.445$: Primitive of $\dfrac 1 {\cot a x}$
- $14.446$: Primitive of $x \cot a x$
- $14.447$: Primitive of $\dfrac {\cot a x} x$
- $14.448$: Primitive of $x \cot^2 a x$
- $14.449$: Primitive of $\dfrac 1 {p + q \cot a x}$
- $14.450$: Primitive of $\cot^n a x$
- $14.451$: Primitive of $\sec a x$
- $14.452$: Primitive of $\sec^2 a x$
- $14.453$: Primitive of $\sec^3 a x$
- $14.454$: Primitive of $\sec^n a x \tan a x$
- $14.455$: Primitive of $\dfrac 1 {\sec a x}$
- $14.456$: Primitive of $x \sec a x$
- $14.457$: Primitive of $\dfrac {\sec a x} x$
- $14.458$: Primitive of $x \sec^2 a x$
- $14.459$: Primitive of $\dfrac 1 {q + p \sec a x}$
- $14.460$: Primitive of $\sec^n a x$
- $14.461$: Primitive of $\csc a x$
- $14.462$: Primitive of $\csc^2 a x$
- $14.463$: Primitive of $\csc^3 a x$
- $14.464$: Primitive of $\csc^n a x \cot a x$
- $14.465$: Primitive of $\dfrac 1 {\csc a x}$
- $14.466$: Primitive of $x \csc a x$
- $14.467$: Primitive of $\dfrac {\csc a x} x$
- $14.468$: Primitive of $x \sec^2 a x$
- $14.469$: Primitive of $\dfrac 1 {q + p \csc a x}$
- $14.470$: Primitive of $\csc^n a x$
- $14.471$: Primitive of $\sin^{-1} \dfrac x a$
- $14.472$: Primitive of $x \sin^{-1} \dfrac x a$
- $14.473$: Primitive of $x^2 \sin^{-1} \dfrac x a$
- $14.474$: Primitive of $\dfrac {\sin^{-1} \dfrac x a} x$
- $14.475$: Primitive of $\dfrac {\sin^{-1} \dfrac x a} {x^2}$
- $14.476$: Primitive of $\left({\sin^{-1} \dfrac x a}\right)^2$
- $14.477$: Primitive of $\cos^{-1} \dfrac x a$
- $14.478$: Primitive of $x \cos^{-1} \dfrac x a$
- $14.479$: Primitive of $x^2 \cos^{-1} \dfrac x a$
- $14.480$: Primitive of $\dfrac {\cos^{-1} \dfrac x a} x$
- $14.481$: Primitive of $\dfrac {\cos^{-1} \dfrac x a} {x^2}$
- $14.482$: Primitive of $\left({\cos^{-1} \dfrac x a}\right)^2$
- $14.483$: Primitive of $\tan^{-1} \dfrac x a$
- $14.484$: Primitive of $x \tan^{-1} \dfrac x a$
- $14.485$: Primitive of $x^2 \tan^{-1} \dfrac x a$
- $14.486$: Primitive of $\dfrac {\tan^{-1} \dfrac x a} x$
- $14.487$: Primitive of $\dfrac {\tan^{-1} \dfrac x a} {x^2}$
- $14.488$: Primitive of $\cot^{-1} \dfrac x a$
- $14.489$: Primitive of $x \cot^{-1} \dfrac x a$
- $14.490$: Primitive of $x^2 \cot^{-1} \dfrac x a$
- $14.491$: Primitive of $\dfrac {\cot^{-1} \dfrac x a} x$
- $14.492$: Primitive of $\dfrac {\cot^{-1} \dfrac x a} {x^2}$
- $14.493$: Primitive of $\sec^{-1} \dfrac x a$
- $14.494$: Primitive of $x \sec^{-1} \dfrac x a$
- $14.495$: Primitive of $x^2 \sec^{-1} \dfrac x a$
- $14.496$: Primitive of $\dfrac {\sec^{-1} \dfrac x a} x$
- $14.497$: Primitive of $\dfrac {\sec^{-1} \dfrac x a} {x^2}$
- $14.498$: Primitive of $\csc^{-1} \dfrac x a$
- $14.499$: Primitive of $x \csc^{-1} \dfrac x a$
- $14.500$: Primitive of $x^2 \csc^{-1} \dfrac x a$
- $14.501$: Primitive of $\dfrac {\csc^{-1} \dfrac x a} x$
- $14.502$: Primitive of $\dfrac {\csc^{-1} \dfrac x a} {x^2}$
- $14.503$: Primitive of $x^m \sin^{-1} \dfrac x a$
- $14.504$: Primitive of $x^m \cos^{-1} \dfrac x a$
- $14.505$: Primitive of $x^m \tan^{-1} \dfrac x a$
- $14.506$: Primitive of $x^m \cot^{-1} \dfrac x a$
- $14.507$: Primitive of $x^m \sec^{-1} \dfrac x a$
- $14.508$: Primitive of $x^m \csc^{-1} \dfrac x a$
- $14.509$: Primitive of $e^{a x}$
- $14.510$: Primitive of $x e^{a x}$
- $14.511$: Primitive of $x^2 e^{a x}$
- $14.512$: Primitive of $x^n e^{a x}$
- $14.513$: Primitive of $\dfrac {e^{a x}} x$
- $14.514$: Primitive of $\dfrac {e^{a x}} {x^n}$
- $14.515$: Primitive of $\dfrac 1 {p + q e^{a x}}$
- $14.516$: Primitive of $\dfrac 1 {\left({p + q e^{a x}}\right)^2}$
- $14.517$: Primitive of $\dfrac 1 {p e^{a x} + q e^{-a x}}$
- $14.518$: Primitive of $e^{a x} \sin b x$
- $14.519$: Primitive of $e^{a x} \cos b x$
- $14.520$: Primitive of $x e^{a x} \sin b x$
- $14.521$: Primitive of $x e^{a x} \cos b x$
- $14.522$: Primitive of $e^{a x} \ln x$
- $14.523$: Primitive of $e^{a x} \sin^n b x$
- $14.524$: Primitive of $e^{a x} \cos^n b x$
- $14.525$: Primitive of $\ln x$
- $14.526$: Primitive of $x \ln x$
- $14.527$: Primitive of $x^m \ln x$
- $14.528$: Primitive of $\dfrac {\ln x} x$
- $14.529$: Primitive of $\dfrac {\ln x} {x^2}$
- $14.530$: Primitive of $\dfrac {\ln x} x$
- $14.531$: Primitive of $\dfrac {\ln^n x} x$
- [If $n = -1$, see $14.532$: Primitive of $\dfrac 1 {x \ln x}$.]
- $14.532$: Primitive of $\dfrac 1 {x \ln x}$
- $14.533$: Primitive of $\dfrac 1 {\ln x}$
- $14.534$: Primitive of $\dfrac {x^m} {\ln x}$
- $14.535$: Primitive of $\ln^n x$
- $14.536$: Primitive of $x^m \ln^n x$
- $14.537$: Primitive of $\ln \left({x^2 + a^2}\right)$
- $14.537$: Primitive of $\ln \left({x^2 - a^2}\right)$
- $14.539$: Primitive of $x^m \ln \left({x^2 \pm a^2}\right)$
- $14.539.1$: Primitive of $x^m \ln \left({x^2 + a^2}\right)$
- $14.539.2$: Primitive of $x^m \ln \left({x^2 - a^2}\right)$
- $14.540$: Primitive of $\sinh a x$
- $14.541$: Primitive of $x \sinh a x$
- $14.542$: Primitive of $x^2 \sinh a x$
- $14.543$: Primitive of $\dfrac {\sinh a x} x$
- $14.544$: Primitive of $\dfrac {\sinh a x} {x^2}$
- $14.545$: Primitive of $\dfrac 1 {\sinh a x}$
- $14.546$: Primitive of $\dfrac x {\sinh a x}$
- $14.547$: Primitive of $\sinh^2 a x$
- $14.548$: Primitive of $x \sinh^2 a x$
- $14.549$: Primitive of $\dfrac 1 {\sinh^2 a x}$
- $14.550$: Primitive of $\sinh a x \sinh p x$
- For $a = \pm p$ see $14.547$: Primitive of $\sinh^2 a x$.
- $14.551$: Primitive of $\sinh a x \sin p x$
- $14.552$: Primitive of $\sinh a x \cos p x$
- $14.553$: Primitive of $\dfrac 1 {p + q \sinh a x}$
- $14.554$: Primitive of $\dfrac 1 {\left({p + q \sinh a x}\right)^2}$
- $14.555$: Primitive of $\dfrac 1 {p^2 + q^2 \sinh^2 a x}$
- $14.556$: Primitive of $\dfrac 1 {p^2 - q^2 \sinh^2 a x}$
- $14.557$: Primitive of $x^m \sinh a x$
- $14.558$: Primitive of $\sinh^n a x$
- $14.559$: Primitive of $\dfrac {\sinh a x} {x^n}$
- $14.560$: Primitive of $\dfrac 1 {\sinh^n a x}$
- $14.561$: Primitive of $\dfrac x {\sinh^n a x}$
- $14.562$: Primitive of $\cosh a x$
- $14.563$: Primitive of $x \cosh a x$
- $14.564$: Primitive of $x^2 \cosh a x$
- $14.565$: Primitive of $\dfrac {\cosh a x} x$
- $14.566$: Primitive of $\dfrac {\cosh a x} {x^2}$
- $14.567$: Primitive of $\dfrac 1 {\cosh a x}$
- $14.568$: Primitive of $\dfrac x {\cosh a x}$
- $14.569$: Primitive of $\cosh^2 a x$
- $14.570$: Primitive of $x \cosh^2 a x$
- $14.571$: Primitive of $\dfrac 1 {\cosh^2 a x}$
- $14.572$: Primitive of $\cosh a x \cosh p x$
- $14.573$: Primitive of $\cosh a x \sin p x$
- $14.574$: Primitive of $\cosh a x \cos p x$
- $14.575$: Primitive of $\dfrac 1 {\cosh a x + 1}$
- $14.576$: Primitive of $\dfrac 1 {\cosh a x - 1}$
- $14.577$: Primitive of $\dfrac x {\cosh a x + 1}$
- $14.578$: Primitive of $\dfrac x {\cosh a x - 1}$
- $14.579$: Primitive of $\dfrac 1 {\left({\cosh a x + 1}\right)^2}$
- $14.580$: Primitive of $\dfrac 1 {\left({\cosh a x - 1}\right)^2}$
- $14.581$: Primitive of $\dfrac 1 {p + q \cosh a x}$
- $14.582$: Primitive of $\dfrac 1 {\left({p + q \cosh a x}\right)^2}$
- $14.583$: Primitive of $\dfrac 1 {p^2 + q^2 \cosh^2 a x}$
- $14.584$: Primitive of $\dfrac 1 {p^2 - q^2 \cosh^2 a x}$
- $14.585$: Primitive of $x^m \cosh a x$
- $14.586$: Primitive of $\cosh^n a x$
- $14.587$: Primitive of $\dfrac {\cosh a x} {x^n}$
- $14.588$: Primitive of $\dfrac 1 {\cosh^n a x}$
- $14.589$: Primitive of $\dfrac x {\cosh^n a x}$
- $14.590$: Primitive of $\sinh a x \cosh a x$
- $14.591$: Primitive of $\sinh p x \cosh q x$
- $14.592$: Primitive of $\sinh^n a x \cosh a x$
- $14.593$: Primitive of $\cosh^n a x \sinh a x$
- $14.594$: Primitive of $\sinh^2 a x \cosh^2 a x$
- $14.595$: Primitive of $\dfrac 1 {\sinh a x \cosh a x}$
- $14.596$: Primitive of $\dfrac 1 {\sinh^2 a x \cosh a x}$
- $14.597$: Primitive of $\dfrac 1 {\sinh a x \cosh^2 a x}$
- $14.598$: Primitive of $\dfrac 1 {\sinh^2 a x \cosh^2 a x}$
- $14.599$: Primitive of $\dfrac {\sinh^2 a x} {\cosh a x}$
- $14.600$: Primitive of $\dfrac {\cosh^2 a x} {\sinh a x}$
- $14.601$: Primitive of $\dfrac 1 {\cosh a x \left({1 + \sinh a x}\right)}$
- $14.602$: Primitive of $\dfrac 1 {\sinh a x \left({\sinh a x + 1}\right)}$
- $14.603$: Primitive of $\dfrac 1 {\sinh a x \left({\sinh a x - 1}\right)}$
- $14.604$: Primitive of $\tanh a x$
- $14.605$: Primitive of $\tanh^2 a x$
- $14.606$: Primitive of $\tanh^3 a x$
- $14.607$: Primitive of $\tanh^n a x \operatorname{sech}^2 a x$
- $14.608$: Primitive of $\dfrac {\operatorname{sech}^2 a x} {\tanh a x}$
- $14.609$: Primitive of $\dfrac 1 {\tanh a x}$
- $14.610$: Primitive of $x \tanh a x$
- $14.611$: Primitive of $\dfrac {\tanh a x} x$
- $14.612$: Primitive of $x \tanh^2 a x$
- $14.613$: Primitive of $\dfrac 1 {p + q \tanh a x}$
- $14.614$: Primitive of $\tanh^n a x$
- $14.615$: Primitive of $\coth a x$
- $14.616$: Primitive of $\coth^2 a x$
- $14.617$: Primitive of $\coth^3 a x$
- $14.618$: Primitive of $\coth^n a x \operatorname{csch}^2 a x$
- $14.619$: Primitive of $\dfrac {\operatorname{csch}^2 a x} {\coth a x}$
- $14.620$: Primitive of $\dfrac 1 {\coth a x}$
- $14.621$: Primitive of $x \coth a x$
- $14.622$: Primitive of $\dfrac {\coth a x} x$
- $14.623$: Primitive of $x \coth^2 a x$
- $14.624$: Primitive of $\dfrac 1 {p + q \coth a x}$
- $14.625$: Primitive of $\coth^n a x$
- $14.626$: Primitive of $\sech a x$
- $14.627$: Primitive of $\sech^2 a x$
- $14.628$: Primitive of $\sech^3 a x$
- $14.629$: Primitive of $\sech^n a x \tanh a x$
- $14.630$: Primitive of $\dfrac 1 {\sech a x}$
- $14.631$: Primitive of $x \sech a x$
- $14.632$: Primitive of $x \sech^2 a x$
- $14.633$: Primitive of $\dfrac {\sech a x} x$
- $14.634$: Primitive of $\dfrac 1 {q + p \sech a x}$
- $14.635$: Primitive of $\sech^n a x$
- $14.636$: Primitive of $\csch a x$
- $14.637$: Primitive of $\csch^2 a x$
- $14.638$: Primitive of $\csch^3 a x$
- $14.639$: Primitive of $\csch^n a x \coth a x$
- $14.640$: Primitive of $\dfrac 1 {\csch a x}$
- $14.641$: Primitive of $x \csch a x$
- $14.642$: Primitive of $x \csch^2 a x$
- $14.643$: Primitive of $\dfrac {\csch a x} x$
- $14.644$: Primitive of $\dfrac 1 {q + p \csch a x}$
- $14.645$: Primitive of $\csch^n a x$
- $14.646$: Primitive of $\arsinh \dfrac x a$
- $14.647$: Primitive of $x \arsinh \dfrac x a$
- $14.648$: Primitive of $x^2 \arsinh \dfrac x a$
- $14.649$: Primitive of $\dfrac 1 x \arsinh \dfrac x a x$
- $14.650$: Primitive of $\dfrac 1 {x^2} \arsinh \dfrac x a$
- $14.651$: Primitive of $\arcosh \dfrac x a$
- $14.652$: Primitive of $x \arcosh \dfrac x a$
- $14.653$: Primitive of $x^2 \arcosh \dfrac x a$
- $14.654$: Primitive of $\dfrac 1 x \arcosh \dfrac x a$
- $14.655$: Primitive of $\dfrac 1 {x^2} \arcosh \dfrac x a$
- $14.656$: Primitive of $\artanh \dfrac x a$
- $14.657$: Primitive of $x \artanh \dfrac x a$
- $14.658$: Primitive of $x^2 \artanh \dfrac x a$
- $14.659$: Primitive of $\dfrac 1 x \artanh \dfrac x a$
- $14.660$: Primitive of $\dfrac 1 {x^2} \artanh \dfrac x a$
- $14.661$: Primitive of $\arcoth \dfrac x a$
- $14.662$: Primitive of $x \arcoth \dfrac x a$
- $14.663$: Primitive of $x^2 \arcoth \dfrac x a$
- $14.664$: Primitive of $\dfrac 1 x \arcoth \dfrac x a$
- $14.665$: Primitive of $\dfrac 1 {x^2} \arcoth \dfrac x a$
- $14.666$: Primitive of $\arsech \dfrac x a$
- $14.667$: Primitive of $x \arsech \dfrac x a$
- $14.668$: Primitive of $\dfrac 1 x \arsech \dfrac x a$
- $14.669$: Primitive of $\arcsch \dfrac x a$
- $14.670$: Primitive of $x \arcsch \dfrac x a$
- $14.671$: Primitive of $\dfrac 1 x \arcsch \dfrac x a$
- $14.672$: Primitive of $x^m \arsinh \dfrac x a$
- $14.673$: Primitive of $x^m \arcosh \dfrac x a$
- $14.674$: Primitive of $x^m \artanh \dfrac x a$
- $14.675$: Primitive of $x^m \arcoth \dfrac x a$
- $14.676$: Primitive of $x^m \arsech \dfrac x a$
- $14.677$: Primitive of $x^m \arcsch \dfrac x a$
Still to be completed
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