Evaluate the definite integrals`int_0^(pi/2)sin2xtan^(-1)(sinx)dx`

Explanation

To evaluate the definite integral \( I = \int_0^{\frac{\pi}{2}} \sin(2x) \tan^{-1}(\sin x) \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
Using the double angle identity for sine, we can rewrite \( \sin(2x) \) as \( 2 \sin x \cos x \):
\[ I = \int_0^{\frac{\pi}{2}} 2 \sin x \cos x \tan^{-1}(\sin x) \, dx \]

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