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The value of the integral
$\int_{0}^{\pi / 2}\left(\sin ^{100} x-\cos ^{100} x\right) d x$ is
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$\int_{0}^{\pi / 2}\left(\sin ^{100} x-\cos ^{100} x\right) d x$ is
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Let $I=\int_{0}^{\pi / 2}\left(\sin ^{100} x-\cos ^{100} x\right) d x$
$$
\begin{aligned}
&=\int_{0}^{\pi / 2} \sin ^{100} x d x-\int_{0}^{\pi / 2} \cos ^{100} x d x \\
&=\left[\frac{(\sin x)^{101}}{101} \cdot \cos x\right]_{0}^{\pi / 2} \\
&-\left[\frac{(\cos x)^{101}}{101}(-\sin x)\right]_{0}^{\pi / 2} \\
&=0+0=0
\end{aligned}
$$
$$
\begin{aligned}
&=\int_{0}^{\pi / 2} \sin ^{100} x d x-\int_{0}^{\pi / 2} \cos ^{100} x d x \\
&=\left[\frac{(\sin x)^{101}}{101} \cdot \cos x\right]_{0}^{\pi / 2} \\
&-\left[\frac{(\cos x)^{101}}{101}(-\sin x)\right]_{0}^{\pi / 2} \\
&=0+0=0
\end{aligned}
$$
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