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