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Q)

Evaluate the following problems using properties of integration: $\int\limits_{0}^{\large\frac{\pi}{2}}\sin^{3}x\cos xdx$

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A)
Toolbox:
  • $\int \limits_{-a}^a f(x) dx=2 \int \limits_0^a f(x) dx $ if f is an even function
  • $\int \limits_{-a}^a f(x) dx=0 $ if f is an odd function
Given $\int\limits_{0}^{\large\frac{\pi}{2}}\sin^{3}x\cos xdx$
Step 1:
$\int\limits_{0}^{\large\frac{\pi}{2}}\sin^{3}x\cos xdx=\large\frac{\sin ^4 x}{4} \bigg]_0^{\large\frac{\pi}{2}}=\large\frac{1}{4}$
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