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# Prove the following$\int\limits_{-1}^1x^{17}\cos^4x\;dx=0$

Can you answer this question?

Toolbox:
• (i) If $f(-x)=f(x)$ then it is an odd function
• (ii) $\int \limits_{-a}^a f(x) =0$ if f(x) is an odd function
Given Prove that $\int \limits_{-1}^1 x^{17} \cos ^4 x dx =0$

Given $I= \int \limits_{-1}^1 x^{17} \cos ^4 x dx =0$

Let $f(x)=x^17 \cos ^4 x$

Therefore $f(-x)=(-x)^{17}\cos ^4 (-x)=-x^{17} \cos ^4x$

$=-f(x)$

Hence the given function is an odd function.

It is known that $\int \limits_{-a}^a f(x)=0$ if f(x) is an odd function

Hence $\int \limits_{-1}^1 x^{17}\cos ^4 xdx=0$

Hence proved

answered Mar 6, 2013 by