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$\int_{-1}^{1} x^{17} \cos ^{4} x d x=0$


Let $I=\int_{-1}^{1} x^{17} \cos ^{4} x d x$

Also, let $f(x)=x^{17} \cos ^{4} x$

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

Therefore, $f(x)$ is an odd function.

It is known that if $f(x)$ is an odd function, then $\int_{-a}^{a} f(x) d x=0$

$\therefore I=\int_{-1}^{1} x^{17} \cos ^{4} x d x=0$

Hence, the given result is proved.

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