Question

The value of the integral $\int\limits_0^{\pi/2}$($Sin^{100} x-Cos^{100}x)dx$ is

• A. $\frac {100!}{(100)^{100}} $
• B. $\frac {1}{100} $
• C. 0
• D. $\frac {\pi}{100} $

Answer 1

Answer: (C)

Let $I =\int\limits_{0}^{\pi / 2}\left(\sin ^{100} x-\cos ^{100} x\right)\, d x$
$=\int\limits_{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} $
$=0+0=0$