Question

The value of the integral $\underset{0}{\overset{\pi /2}{\int }}$($Si{n}^{100}x-Co{s}^{100}x)dx$ is

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

Answer 1

Answer: (C)

Let $I=\underset{0}{\overset{\pi /2}{\int }}\left({\sin }^{100}x-{\cos }^{100}x\right)dx$
$=\underset{0}{\overset{\pi /2}{\int }}{\sin }^{100}xdx-{\int }_0^{\pi /2}{\cos }^{100}xdx$
$={\left[\frac{(\sin x{)}^{101}}{101}\cdot \cos x\right]}_0^{\pi /2}$
$-{\left[\frac{(\cos x{)}^{101}}{101}(-\sin x)\right]}_0^{\pi 2}$
$=0+0=0$