A) \[\frac{1}{100}\]
B) \[\frac{100!}{{{(100)}^{100}}}\]
C) \[\frac{\pi }{100}\]
D) \[0\]
Correct Answer: D
Solution :
Let \[I=\int_{0}^{\pi /2}{({{\sin }^{100}}x-{{\cos }^{100}}x)dx}\] \[=\int_{0}^{\pi /2}{{{\sin }^{100}}x\,dx-\int_{0}^{\pi /2}{{{\cos }^{100}}}x\,dx}\] \[=\left[ \frac{{{(\sin x)}^{101}}}{101}.\cos x \right]_{0}^{\pi /2}-\left[ \frac{{{(\cos x)}^{101}}}{101}(-\sin x) \right]_{0}^{\pi /2}\]\[=0+0\] \[=0\]
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