A) 1
B) 0
C) \[\frac{1}{30}\]
D) \[\frac{1}{5}\]
Correct Answer: C
Solution :
Let \[I=\int_{0}^{1}{x{{(1-x)}^{4}}dx}\] \[=\int_{0}^{1}{(1-x){{(1-(1-x))}^{4}}dx}\] \[=\int_{0}^{1}{(1-x){{x}^{4}}dx}\] \[=\int_{0}^{1}{({{x}^{4}}-{{x}^{5}})dx}\] \[=\left[ \frac{{{x}^{5}}}{5}-\frac{{{x}^{6}}}{6} \right]_{0}^{1}=\left[ \frac{1}{5}-\frac{1}{6} \right]\] \[=\frac{1}{30}\] Alternate Solution: Let \[I=\int_{0}^{1}{x{{(1-x)}^{4}}dx}\] \[=\left[ -x\frac{{{(1-5)}^{5}}}{5} \right]_{0}^{1}-\int_{0}^{1}{-\frac{{{(1-x)}^{5}}}{5}}dx\] \[=0+\frac{1}{5}\left[ -\frac{{{(1-x)}^{6}}}{6} \right]_{0}^{1}\] \[=-\frac{1}{5}\left( 0-\frac{1}{6} \right)=\frac{1}{30}\]
You need to login to perform this action.
You will be redirected in
3 sec
