Mathematics

The value of $$\displaystyle\int_{-\pi}^{\pi}\sin^{3}x \cos^{2}x\ dx$$ is equal to


ANSWER

$$0$$


SOLUTION
applying, $$\displaystyle \int_{a}^{b}+(x)dx = \int_{a}^{b}f(a+b-x)dx...$$ 
$$\displaystyle \int_{-\pi }^{\pi } sin^{3}x\,cos^{2}dx =  \int_{-\pi }^{\pi } sin^{3}x(1-sin^{2}x)dx $$ 
$$\displaystyle\Rightarrow  \int_{-\pi }^{\pi }(sin^{3}x-sin^{5}x)dx = I....(1) $$
$$\displaystyle \Rightarrow I = \int_{-\pi }^{\pi }(sin^{3}(\pi -\pi -x)-sin^{5}x(\pi -\pi -x))dx $$
$$\displaystyle  I = \int_{-\pi }^{\pi } (-sin^{3}x+sin^{5}x)dx....(2) $$
$$ (1)+(2) \Rightarrow 2\pi = 0 $$
$$ \boxed {I = 0} $$ 
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Single Correct Medium Published on 17th 09, 2020
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