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int(0)^(pi//2)(sinx)/((1+cos^(2)x))dx...

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

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To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1 + \cos^2 x} \, dx \), we will use the substitution method. ### Step 1: Substitution Let \( t = \cos x \). Then, the differential \( dx \) can be expressed in terms of \( dt \): \[ dx = -\frac{1}{\sin x} dt \] Since \( \sin^2 x + \cos^2 x = 1 \), we have \( \sin x = \sqrt{1 - t^2} \). Therefore, we can rewrite \( dx \) as: ...
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