Find the value of `int_(0)^(pi//2)cos^(5)x sin ^(7)xdx`

Explanation

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \cos^5 x \sin^7 x \, dx \), we can use the substitution method and properties of definite integrals. Here’s a step-by-step solution:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int_{0}^{\frac{\pi}{2}} \cos^5 x \sin^7 x \, dx \]
We can express \( \cos^5 x \) in terms of \( \sin x \):
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