Primitive of Power of x by Arctangent of a x
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Theorem
- $\ds \int x^n \arctan a x \rd x = \frac {x^{n + 1} } {n + 1} \arctan a x + \frac a {n + 1} \int \frac {x^{n + 1} \rd x} {a^2 x^2 + 1}$
for $n \ne -1$.
Proof
Recall:
\(\text {(1)}: \quad\) | \(\ds \int x^n \arctan x \rd x\) | \(=\) | \(\ds \frac {x^{n + 1} } {n + 1} \arctan x - \frac 1 {n + 1} \int \frac {x^{n + 1} \rd x} {x^2 + 1}\) | Primitive of $x^n \arctan x$ |
Then:
\(\ds \int x^n \arctan a x \rd x\) | \(=\) | \(\ds \int \dfrac 1 {a^n} \paren {a x}^n \arctan a x \rd x\) | manipulating into appropriate form | |||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {a^n} \int \paren {a x}^n \arctan a x \rd x\) | Primitive of Constant Multiple of Function | |||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {a^n} \paren {\dfrac 1 a \paren {\frac {\paren {a x}^{n + 1} } {n + 1} \arctan a x + \frac 1 {n + 1} \int \frac {\paren {a x}^{n + 1} \rd x} {\paren {a x}^2 + 1} } }\) | Primitive of Function of Constant Multiple, from $(1)$ | |||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^{n + 1} } {n + 1} \arctan a x + \frac a {n + 1} \int \frac {x^{n + 1} \rd x} {a^2 x^2 + 1}\) | simplifying
|
$\blacksquare$
Also see
Sources
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $101$.