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$\int_{0}^{π} {(π x - x^{2})}^{100} \sin 2 x d x$ is equal to

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$π^{100}$

$\frac{1}{2} (π^{100} - π^{97})$

$\frac{1}{2} (π^{100} + π^{97})$

answer is D.

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Detailed Solution

Let $I = \int_{0}^{π} {(π x - x^{2})}^{100} \sin 2 x d x .$ then

$\begin{array}{l} I = \int_{0}^{π} {\{π (π - x) - (π - x)^{2}\}}^{100} \sin 2 (π - x) d x \\ \Rightarrow I = - \int_{0}^{π} {(π x - x^{2})}^{100} \sin 2 x d x \\ \Rightarrow I = - I \Rightarrow 2 I = 0 \Rightarrow I = 0 \end{array}$

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