Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 15

Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 15

Published $\text {1968}$

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$15 \quad$ Definite Integrals

Definition of a Definite Integral

$15.1$: Darboux Integral

$15.2$: Fundamental Theorem of Calculus

$15.3$: Improper Integral on Closed Interval Unbounded Above

$15.4$: Improper Integral on Interval Unbounded Above and Below

$15.5$: Improper Integral on Open Above Interval

$15.6$: Improper Integral on Open Below Interval

General Formulas involving Definite Integrals

$15.7$: Linear Combination of Integrals: Definite Case

$15.8$: Definite Integral of Constant Multiple of Real Function

$15.9$: Definite Integral on Zero Interval

$15.10$: Reversal of Limits of Definite Integral

$15.11$: Sum of Integrals of Adjacent Intervals for Integrable Functions

$15.12$: Mean Value Theorem for Integrals

$15.13$: Mean Value Theorem for Integrals: Generalization

Leibnitz's Rule for Differentiation of Integrals

$15.14$: Leibniz's Integral Rule

Approximate Formulas for Definite Integrals

$15.15$: Rectangular Formula for Definite Integrals

$15.16$: Trapezoidal Formula for Definite Integrals

$15.17$: Simpson's Rule

Definite Integrals involving Rational or Irrational Expressions

$15.18$ Definite Integral to Infinity of $\dfrac 1 {x^2 + a^2}$

$15.19$: Definite Integral to Infinity of $\dfrac {x^{p - 1} } {1 + x}$

$15.20$: Definite Integral to Infinity of $\dfrac {x^m} {x^n + a^n}$

$15.21$: Definite Integral to Infinity of $\dfrac {x^m} {1 + 2 x \cos \beta + x^2}$

$15.22$: Definite Integral from $0$ to $a$ of $\dfrac 1 {\sqrt {a^2 - x^2} }$

$15.23$: Definite Integral from $0$ to $a$ of $\sqrt {a^2 - x^2}$

$15.24$: Definite Integral from $0$ to $a$ of $x^m \paren {a^n - x^n}^p$

$15.25$: Definite Integral from $0$ to Infinity of $\dfrac {x^m} {\paren {x^n + a^n}^r}$

Definite Integrals involving Trigonometric Functions

$15.26$: Definite Integral from $0$ to $\pi$ of $\sin m x \sin n x$

$15.27$: Definite Integral from $0$ to $\pi$ of $\cos m x \cos n x$

$15.28$: Definite Integral from $0$ to $\pi$ of $\sin m x \cos n x$

$15.29$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\sin^2 x$ and $\cos^2 x$

$15.29.1$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\sin^2 x$

$15.29.2$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\cos^2 x$

$15.30$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\sin^{2 n} x$ and $\cos^{2 n} x$

$15.30.1$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\sin^{2 n} x$

$15.30.2$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\cos^{2 n} x$

$15.31$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\sin^{2 n + 1} x$ and $\cos^{2 n + 1} x$

$15.31.1$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\sin^{2 n + 1} x$

$15.31.2$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\cos^{2 n + 1} x$

$15.32$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\sin^{2 p - 1} x \cos^{2 p - 1} x$

$15.33$: Definite Integral to Infinity of $\dfrac {\sin p x} x$

$15.34$: Definite Integral to Infinity of $\dfrac {\sin p x \cos q x} x$

$15.35$: Definite Integral to Infinity of $\dfrac {\sin p x \sin q x} {x^2}$

$15.36$: Definite Integral to Infinity of $\paren {\dfrac {\sin p x} x}^2$

$15.37$: Definite Integral to Infinity of $\dfrac {1 - \cos p x} {x^2}$

$15.38$: Definite Integral to Infinity of $\dfrac {\cos p x - \cos q x} x$

$15.39$: Definite Integral to Infinity of $\dfrac {\cos p x - \cos q x} {x^2}$

$15.40$: Definite Integral to Infinity of $\dfrac {\cos m x} {x^2 + a^2}$

$15.41$: Definite Integral to Infinity of $\dfrac {x \sin m x} {x^2 + a^2}$

$15.42$: Definite Integral to Infinity of $\dfrac {\sin m x} {x \paren {x^2 + a^2} }$

$15.43$: Definite Integral from $0$ to $2 \pi$ of $\dfrac 1 {a + b \sin x}$

$15.44$: Definite Integral from $0$ to $2 \pi$ of $\dfrac 1 {a + b \cos x}$

$15.45$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\dfrac 1 {a + b \cos x}$

$15.46$: Definite Integral from 0 to $2 \pi$ of $\dfrac 1 {\paren {a + b \sin x}^2}$ and $\dfrac 1 {\paren {a + b \cos x}^2}$

$15.46.1$: Definite Integral from $0$ to $2 \pi$ of $\dfrac 1 {\paren {a + b \sin x}^2}$

$15.46.2$: Definite Integral from $0$ to $2 \pi$ of $\dfrac 1 {\paren {a + b \cos x}^2}$

$15.47$: Definite Integral from $0$ to $2 \pi$ of $\dfrac 1 {1 - 2 a \cos x + a^2}$

$15.48$: Definite Integral from $0$ to $2 \pi$ of $\dfrac {x \sin x} {1 - 2 a \cos x + a^2}$

$15.49$: Definite Integral from $0$ to $\pi$ of $\dfrac {\cos m x} {1 - 2 a \cos x + a^2}$

$15.50$: Definite Integral to Infinity of $\sin a x^2$ and $\cos a x^2$

$15.50.1$: Definite Integral to Infinity of $\sin a x^2$

$15.50.2$: Definite Integral to infinity of $\cos a x^2$

$15.51$: Definite Integral to Infinity of $\sin a x^n$

$15.52$: Definite Integral to Infinity of $\cos a x^n$

$15.53$: Definite Integral to Infinity of $\dfrac {\sin x} {\sqrt x}$ and $\dfrac {\cos x} {\sqrt x}$

$15.53.1$: Definite Integral to Infinity of $\dfrac {\sin x} {\sqrt x}$

$15.53.2$: Definite Integral to Infinity of $\dfrac {\cos x} {\sqrt x}$

$15.54$: Definite Integral to Infinity of $\dfrac {\sin x} {x^p}$

$15.55$: Definite Integral to Infinity of $\dfrac {\cos x} {x^p}$

$15.56$: Definite Integral to Infinity of $\sin a x^2 \cos 2 b x$

$15.57$: Definite Integral to Infinity of $\cos a x^2 \cos 2 b x$

$15.58$: Definite Integral to Infinity of $\dfrac {\sin^3 x} {x^3}$

$15.59$: Definite Integral to Infinity of $\dfrac {\sin^4 x} {x^4}$

$15.60$: Definite Integral to Infinity of $\dfrac {\tan x} x$

$15.61$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\dfrac 1 {1 + \tan^m x}$

$15.62$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\dfrac x {\sin x}$

$15.63$: Definite Integral from $0$ to $1$ of $\dfrac {\arctan x} x$

$15.64$: Definite Integral from $0$ to $1$ of $\dfrac {\arcsin x} x$

$15.65$: Euler-Mascheroni Constant as Difference of Integrals involving Cosine

$15.66$: Definite Integral to Infinity of $\paren {\dfrac 1 {1 + x^2} - \cos x} \dfrac 1 x$

$15.67$: Definite Integral to Infinity of $\dfrac {\arctan p x - \arctan q x} x$

Definite Integrals involving Exponential Functions

$15.68$: Definite Integral to Infinity of $e^{-a x} \cos b x$

$15.69$: Definite Integral to Infinity of $e^{-a x} \sin b x$

$15.70$: Definite Integral to Infinity of $\dfrac {e^{-a x} \sin b x} x$

$15.71$: Definite Integral to Infinity of $\dfrac {e^{-a x} - e^{-b x} } x$

$15.72$: Definite Integral to Infinity of $e^{-a x^2}$

$15.73$: Definite Integral to Infinity of $e^{-a x^2} \cos b x$

$15.74$: Definite Integral to Infinity of $\map \exp {-\paren {a x^2 + b x + c} }$

$15.75$: Definite Integral over Reals of $\map \exp {-\paren {a x^2 + b x + c} }$

$15.76$: Definite Integral to Infinity of $x^n e^{-a x}$

$15.77$: Definite Integral to Infinity of $x^m e^{-a x^2}$

$15.78$: Definite Integral to Infinity of $\map \exp {a x^2 + \dfrac b {x^2} }$

$15.79$: Definite Integral to Infinity of $\dfrac x {e^x - 1}$

$15.80$: Definite Integral to Infinity of $\dfrac {x^{n - 1} } {e^x - 1}$

$15.81$: Definite Integral to Infinity of $\dfrac x {e^x + 1}$

$15.82$: Definite Integral to Infinity of $\dfrac {x^{n - 1} } {e^x + 1}$

$15.83$: Definite Integral to Infinity of $\dfrac {\sin m x} {e^{2 \pi x} - 1}$

$15.84$: Definite Integral to Infinity of $\paren {\dfrac 1 {1 + x} - e^{-x} } \dfrac 1 x$

$15.85$: Definite Integral to Infinity of $\dfrac {e^{-x^2} - e^{-x} } x$

$15.86$: Definite Integral to Infinity of $\dfrac 1 {e^x - 1} - \dfrac {e^{-x} } x$

$15.87$: Definite Integral to Infinity of $\dfrac {e^{-a x} - e^{-b x} } {x \sec p x}$

$15.88$: Definite Integral to Infinity of $\dfrac {e^{-a x} - e^{-b x} } {x \csc p x}$

$15.89$: Definite Integral to Infinity of $\dfrac {e^{-a x} \paren {1 - \cos x} } {x^2}$

Definite Integrals involving Logarithmic Functions

$15.90$: Definite Integral from $0$ to $1$ of $x^m \paren {\ln x}^n$

$15.91$: Definite Integral from $0$ to $1$ of $\dfrac {\ln x} {1 + x}$

$15.92$: Definite Integral from $0$ to $1$ of $\dfrac {\ln x} {1 - x}$

$15.93$: Definite Integral from $0$ to $1$ of $\dfrac {\map \ln {1 + x} } x$

$15.94$: Definite Integral from $0$ to $1$ of $\dfrac {\map \ln {1 - x} } x$

$15.95$: Definite Integral from $0$ to $1$ of $\ln x \map \ln {1 + x}$

$15.96$: Definite Integral from $0$ to $1$ of $\ln x \map \ln {1 - x}$

$15.97$: Definite Integral to Infinity of $\dfrac {x^{p - 1} \ln x} {1 + x}$

$15.98$: Definite Integral from $0$ to $1$ of $\dfrac {x^m - x^n} {\ln x}$

$15.99$: Definite Integral to Infinity of $e^{-x} \ln x$

$15.100$: Definite Integral to Infinity of $e^{-x^2} \ln x$

$15.101$: Definite Integral to Infinity of $\map \ln {\dfrac {e^x + 1} {e^x - 1} }$

$15.102$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\map \ln {\sin x}$ and $\map \ln {\cos x}$

$15.102.1$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\map \ln {\sin x}$

$15.102.2$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\map \ln {\cos x}$

$15.103$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\paren {\map \ln {\sin x}}^2$ and $\paren {\map \ln {\cos x}}^2$

$15.103.1$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\paren {\map \ln {\sin x}}^2$

$15.103.2$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\paren {\map \ln {\cos x}}^2$

$15.104$: Definite Integral from $0$ to $\pi$ of $x \map \ln {\sin x}$

$15.105$: Definite Integral from $0$ to $\dfrac \pi 2$ of $\sin x \map \ln {\sin x}$

$15.106$: Definite Integral from $0$ to $2 \pi$ of $\map \ln {a + b \sin x}$ and $\map \ln {a + b \cos x}$

$15.106.1$: Definite Integral from $0$ to $2 \pi$ of $\map \ln {a + b \sin x}$

$15.106.2$: Definite Integral from $0$ to $2 \pi$ of $\map \ln {a + b \cos x}$

$15.107$: Definite Integral from $0$ to $\pi$ of $\map \ln {a + b \cos x}$

$15.108$: Definite Integral from $0$ to $\pi$ of $\map \ln {a^2 - 2 a b \cos x + b^2}$

$15.109$: Definite Integral from $0$ to $\dfrac \pi 4$ of $\map \ln {1 + \tan x}$

$15.110$: Definite Integral from $0$ to $\pi$ of $\sec x \map \ln {\dfrac {1 + b \cos x} {1 + a \cos x} }$

$15.111$: Definite Integral from $0$ to $a$ of $\map \ln {2 \sin \dfrac x 2}$

Definite Integrals involving Hyperbolic Functions

$15.112$: Definite Integral to Infinity of $\dfrac {\sin a x} {\sinh b x}$

$15.113$: Definite Integral to Infinity of $\dfrac {\cos a x} {\cosh b x}$

$15.114$: Definite Integral to Infinity of $\dfrac x {\sinh a x}$

$15.115$: Definite Integral to Infinity of $\dfrac {x^n} {\sinh a x}$

$15.116$: Definite Integral to Infinity of $\dfrac {\sinh a x} {e^{b x} + 1}$

$15.117$: Definite Integral to Infinity of $\dfrac {\sinh a x} {e^{b x} -1}$

Miscellaneous Definite Integrals

$15.118$: Frullani's Integral

$15.119$: Definite Integral from $0$ to $1$ of $x^{-x}$

$15.120$: Definite Integral from $-a$ to $a$ of $\paren {a + x}^{m - 1} \paren {a - x}^{n - 1}$

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