Primitive of Arcsine Function/Proof 1
Theorem
- $\ds \int \arcsin x \rd x = x \arcsin x + \sqrt {1 - x^2} + C$
Proof
Let:
| \(\ds u\) | \(=\) | \(\ds \arcsin x\) | ||||||||||||
| \(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \sin u\) | \(=\) | \(\ds x\) | Definition of Real Arcsine | |||||||||
| \(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \cos u\) | \(=\) | \(\ds \sqrt {1 - x^2}\) | Sum of Squares of Sine and Cosine |
Then:
| \(\ds \int \arcsin x \rd x\) | \(=\) | \(\ds \int u \cos u \rd u\) | Primitive of Function of Arcsine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos u + u \sin u + C\) | Primitive of $x \cos a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos u + u x + C\) | Substitution for $\sin u$ from $\paren 1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {1 - x^2} + u x + C\) | Substitution for $\cos u$ from $\paren 2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arcsin x + \sqrt {1 - x^2} + C\) | Substitution for $u$ and rearranging |
$\blacksquare$