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Mathematics Question on Some Properties of Definite Integrals

The value of the integral ∫0π/2\int\limits_0^{\pi/2}(Sin100x−Cos100x)dxSin^{100} x-Cos^{100}x)dx is

A

100!(100)100\frac {100!}{(100)^{100}}

B

1100\frac {1}{100}

C

0

D

π100\frac {\pi}{100}

Answer

0

Explanation

Solution

Let I=∫0π/2(sin⁡100x−cos⁡100x) dxI =\int\limits_{0}^{\pi / 2}\left(\sin ^{100} x-\cos ^{100} x\right)\, d x
=∫0π/2sin⁡100xdx−∫0π/2cos⁡100 x dx=\int\limits_{0}^{\pi / 2} \sin ^{100} x d x-\int_{0}^{\pi / 2} \cos ^{100}\, x \,d x
=[(sin⁡x)101101⋅cos⁡x]0π/2=\left[\frac{(\sin x)^{101}}{101} \cdot \cos x\right]_{0}^{\pi / 2}
−[(cos⁡x)101101(−sin⁡x)]0π2-\left[\frac{(\cos x)^{101}}{101}(-\sin x)\right]_{0}^{\pi 2}
=0+0=0=0+0=0