The value of $\int_{-\pi }^{\pi } \sin ^{3} x \cos ^{2} x \: dx=... ?$

 For finding the value of $\int_{-\pi }^{\pi } \sin ^{3} x \cos ^{2} x \: \mathbf{dx}$ :

Consider the property of definite integral 

$\mathbf{\int_{-a}^{a} f(x) dx} =\left\{\begin{matrix} 2\int_{0}^{a} f(x)dx \; \; if\; \textbf{f is even}, f(-x)=f(x)\\ 0 \;\; \; \; \; \; \; \; \;\; \; \; \; \; \; \: if \: \textbf{ f is odd},f(-x)=-f(x) \end{matrix}\right.$ 

Here we have $f(x)=\sin ^{3} x \cos ^{2} x$ 

And  \[\begin{align*} f(-x) &= \sin ^{3}(-x) \cos ^{2}(-x) \\ &= (\sin (\, -x))^{3}(\cos (\, -x))^{2} \\ &= (-\sin x)^{3} (\cos x)^{2} \; \; [\because \sin (-x)=-\sin x \; and\; \cos (-x)=\cos x]\\ &= -\sin ^{3} x \cos ^{2} x\\ &= -f(x) \end{align*}\]

Hence $f(x)=\sin ^{3} x \cos ^{2} x$  is a odd function.

And 

\[\int_{-\pi }^{\pi } f(x) \; dx = \int_{-\pi }^{\pi } \sin ^{3}x \cos ^{2} x\; dx=0\]

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