Integral of Arcsine Function
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 It has been suggested that this page or section be merged into Primitive of Arcsine of x over a. (Discuss) |
Theorem
- $\displaystyle \int \arcsin x \rd x = x \arcsin x + \sqrt {1 - x^2} + C$
for $x \in \closedint {-1} 1$.
Proof
| \(\displaystyle \int \arcsin x \rd x\) | \(=\) | \(\displaystyle \int 1 \cdot \arcsin x \rd x\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle x \arcsin x - \int x \paren {\frac \rd {\rd x} \arcsin x} \rd x\) | Integration by Parts | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle x \arcsin x - \int \frac x {\sqrt {1 - x^2} } \rd x\) | Derivative of Arcsine Function |
| \(\displaystyle u\) | \(=\) | \(\displaystyle 1 - x^2\) | |||||||||||
| \(\displaystyle \frac {\d u} {\d x}\) | \(=\) | \(\displaystyle -2 x\) | differentiating with respect to $x$ | ||||||||||
| \(\displaystyle \leadsto \ \ \) | \(\displaystyle -\frac 1 2 \frac {\d u} {\d x}\) | \(=\) | \(\displaystyle x\) | ||||||||||
| \(\displaystyle \int \arcsin x \rd x\) | \(=\) | \(\displaystyle x \arcsin x - \int \frac {- \frac 1 2} {\sqrt u} \frac {\d u} {\d x} \rd x\) | |||||||||||
| \(\displaystyle \int \arcsin x \rd x\) | \(=\) | \(\displaystyle x \arcsin x + \frac 1 2 \int u^{-1/2} \rd u\) | Integration by Substitution | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle x \arcsin x + u^{1/2} + C\) | Integral of Power | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle x \arcsin x + \sqrt {1 - x^2} + C\) | from $u = 1 - x^2$ |
$\blacksquare$
Sources
- Weisstein, Eric W. "Inverse Sine." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseSine.html
