Prove the following Exercise: ^ 1 _ -1 x^ 17 ^ 4 x dx=0 - Vidyadip

Question

Prove the following Exercise:
$\int^{1}_{-1}\text{x}^{17}\cos^{4}\text{x}\ \text{dx}=0$

✓

Answer

$\text{Let I}=\int^{1}\limits_{-1}\text{x}^{17}\cos^{4}\text{x dx}$
Also, let $\text{f(x)}=\text{x}^{17}\cos^{4}\text{x}$
$\Rightarrow\text{f}\ (-\text{x)}=(-\text{x)}^{17}\cos^{4}(-\text{x)}=-\text{x}^{17}\cos^{4}\text{x}=-\text{f (x)}$
Therefore, f(x) is an odd function.
it is know that if f(x) is an odd function, then $\int^{\text{a}}\limits_{-\text{a}}\text{f (x) dx}=0$
$\therefore\text{I}=\int^{1}\limits_{-1}\text{x}^{17}\cos^{4}\text{x dx}=0$
Hence, the given result is Proved.

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