Evaluate : `int " sin"^(-1) sqrt((x)/(a+x)) dx `

Explanation

To evaluate the integral \( I = \int \sin^{-1} \left( \sqrt{\frac{x}{a+x}} \right) dx \), we will use the substitution method and integration by parts. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( x = a \tan^2 \theta \). Then, we differentiate to find \( dx \):
\[ dx = 2a \tan \theta \sec^2 \theta \, d\theta \]

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