Find `int(xsin^(-1)x)/(sqrt(1-x^2))dx`

Explanation

To solve the integral \( \int \frac{x \sin^{-1}(x)}{\sqrt{1 - x^2}} \, dx \), we will use the substitution method followed by integration by parts. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = \sin^{-1}(x) \). Then, we differentiate both sides with respect to \( x \):
\[ \frac{dt}{dx} = \frac{1}{\sqrt{1 - x^2}} \implies dt = \frac{1}{\sqrt{1 - x^2}} \, dx \implies dx = \sqrt{1 - x^2} \, dt \]
Also, since \( x = \sin(t) \), we can substitute \( x \) in terms of \( t \).
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