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. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mergeto}} from the code. |
Theorem
- $\ds \int \arcsin x \rd x = x \arcsin x + \sqrt {1 - x^2} + C$
for $x \in \closedint {-1} 1$.
Proof
| \(\ds \int \arcsin x \rd x\) | \(=\) | \(\ds \int 1 \cdot \arcsin x \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds x \arcsin x - \int x \paren {\frac \rd {\rd x} \arcsin x} \rd x\) | Integration by Parts | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arcsin x - \int \frac x {\sqrt {1 - x^2} } \rd x\) | Derivative of Arcsine Function |
| \(\ds u\) | \(=\) | \(\ds 1 - x^2\) | ||||||||||||
| \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds -2 x\) | differentiating with respect to $x$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds -\frac 1 2 \frac {\d u} {\d x}\) | \(=\) | \(\ds x\) | |||||||||||
| \(\ds \int \arcsin x \rd x\) | \(=\) | \(\ds x \arcsin x - \int \frac {- \frac 1 2} {\sqrt u} \frac {\d u} {\d x} \rd x\) | Integration by Substitution | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arcsin x + \frac 1 2 \int u^{-1/2} \rd u\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds x \arcsin x + u^{1/2} + C\) | Integral of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arcsin x + \sqrt {1 - x^2} + C\) | from $u = 1 - x^2$ |
$\blacksquare$
Sources
- Weisstein, Eric W. "Inverse Sine." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseSine.html