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

Explanation

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1 + \cos^2 x} \, dx \), we can use a substitution method. Here’s the step-by-step solution:

### Step 1: Substitution
Let \( \cos x = t \). 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 \( \sin x = \sqrt{1 - t^2} \).
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