Prove _ -1 ^ 1 x^ 17 ^ 4 x d x=0 - Vidyadip

Question

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

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Answer

Let $\mathrm{I}=\int_{-1}^{1} \mathrm{x}^{17} \cdot \cos ^{4} \mathrm{xdx}$
As we can see f(x) =x¹⁷.cos⁴x and f(-x) = (-x)¹⁷.cos⁴(-x) = -x¹⁷.cos⁴x
i.e. f(x) = -f(-x)
so, it is an odd function.
It is also known that if f(x) is an odd function then $\left\{\int_{-a}^{a} f(x) d x=0\right\}$
$\Rightarrow \mathrm{I}=\int_{-1}^{1} \mathrm{x}^{17} \cdot \cos ^{4} \mathrm{xdx}=0$
Hence proved.

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