Q102 / 180 · KCET 2007 · Mathematics

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
    $\frac{\pi}{100}$
  • Correct answer
    D
    0

Solution

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

Hence, the correct answer is option: 'D'.

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