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Q)

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

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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 $\int\limits_{\large\frac{-\pi}{2}}^{\large\frac{\pi}{2}}\sin^{2}x\cos x dx$
Step 1:
$f(x) =\sin ^2 x \cos x$
$f(-x) =\sin ^2 (-x) \cos (-x)$
$\qquad=\sin ^2 x \cos x=f(x)$
Step 2:
$f(x)$ is an even function
$\int\limits_{\large\frac{-\pi}{2}}^{\large\frac{\pi}{2}}\sin^{2}x\cos x dx=2\int\limits_0^{\large\frac{\pi}{2}}\sin^{2}x\cos x dx$
Step 3:
$\qquad= \large\frac{2 \sin ^3x }{3} \bigg]_0^{\large\frac{\pi}{2}}$
$\qquad=\large\frac{2}{3}$
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