KCET · Maths · Definite Integration
The value of the integral
\(\int_{0}^{\pi / 2}\left(\sin ^{100} x-\cos ^{100} x\right) d x\) is
- A \(\frac{1}{100}\)
- B \(\frac{100 !}{(100)^{100}}\)
- C \(\frac{\pi}{100}\)
- D 0
Answer & Solution
Correct Answer
(D) 0
Step-by-step Solution
Detailed explanation
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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