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Home  >>  CBSE XII  >>  Math  >>  Integrals
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The value of $\large\int\limits_{-\pi}^\pi$$\sin^3x\cos^2x dx$ is___________.

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Toolbox:
  • If $f(-x)=-f(x)$,then $f(x)$ is an odd function.
  • If $f(-x)=f(x)$,then $f(x)$ is an even function.
  • $\int_{-a}^a f(x)dx=0$ if $f(x)$ is an odd function and $\int_{-a}^a f(x)dx=2\int_0^af(x)dx$ if $f(x)$ is an even function.$
Step 1:
Let $I=\int_{\pi}^\pi\sin^3x\cos^2x dx$
First let us check if the given function is odd or even.
Let us replace $x$ by $-x$.
$\sin^3x\cos^3x$ when replaced by $-x$ will become $\sin^3(-x)\cos^2(-x)$
$\;\;=(-\sin^3x).(-\cos^2x)$
Step 2:
$I=-\sin^3x.\cos^2x$
Because $\sin(-x)=-\sin x$ and $\cos(-x)=\cos x$.
Hence the given function is an odd function .
Hence $I=0$.
answered Apr 25, 2013 by sreemathi.v
edited Apr 25, 2013 by sreemathi.v
 
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