Primitive of Power of x by Arcsine of x
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Theorem
- $\ds \int x^m \arcsin x \rd x = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$
Proof
With a view to expressing the primitive in the form:
- $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) | \(=\) | \(\ds \arcsin x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac 1 {\sqrt {1 - x^2} }\) | Derivative of $\arcsin x$ |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds x^m\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1}\) | Primitive of Power |
Then:
\(\ds \int x^m \arcsin x \rd x\) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1} \arcsin x - \int \frac {x^{m + 1} } {m + 1} \paren {\frac 1 {\sqrt {1 - x^2} } } \rd x\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }\) | Primitive of Constant Multiple of Function |
$\blacksquare$