Question

integrate sin^-1 x / sqrt (1-x^2)

Answer 1

$\frac{(\sin^{-1} x)^2}{2} + C$

Answer 2

To integrate the given function, we use the method of substitution.

Let $u = \sin^{-1} x$.

Differentiating $u$ with respect to $x$, we get:

$$\frac{du}{dx} = \frac{1}{\sqrt{1-x^2}}$$

This implies:

$$du = \frac{1}{\sqrt{1-x^2}} dx$$

Now, substitute $u$ and $du$ into the integral:

$$\int \frac{\sin^{-1} x}{\sqrt{1-x^2}} dx = \int u \, du$$

This is a standard integral:

$$\int u \, du = \frac{u^2}{2} + C$$

Finally, substitute back $u = \sin^{-1} x$:

$$\frac{(\sin^{-1} x)^2}{2} + C$$