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int(0)^(pi)(x sinx)/((1+sinx))dx=pi((pi)...

`int_(0)^(pi)(x sinx)/((1+sinx))dx=pi((pi)/(2)-1)`

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To solve the integral \( I = \int_{0}^{\pi} \frac{x \sin x}{1 + \sin x} \, dx \) and prove that it equals \( \pi \left( \frac{\pi}{2} - 1 \right) \), we can use the property of definite integrals. Here’s a step-by-step solution: ### Step 1: Set up the integral Let \[ I = \int_{0}^{\pi} \frac{x \sin x}{1 + \sin x} \, dx \] ...
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