A) \[\frac{1}{10100}\]
B) \[\frac{11}{10100}\]
C) \[\frac{1}{10010}\]
D) \[\frac{11}{11100}\]
E) \[\frac{1}{1010}\]
Correct Answer: A
Solution :
Let\[I=\int_{0}^{1}{x}{{(1-x)}^{99}}dx\] Put \[1-x=t\Rightarrow -dx=dt\] \[\therefore \] \[I=\int_{1}^{0}{(1-t)}{{t}^{99}}dt\] \[=\int_{0}^{1}{[{{t}^{99}}-{{t}^{100}}]}dt=\left[ \frac{{{t}^{100}}}{100}-\frac{{{t}^{101}}}{101} \right]_{0}^{1}\] \[=\frac{1}{100}-\frac{1}{101}=\frac{1}{10100}\]
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