A) \[\frac{58}{55}\]
B) \[\frac{55}{58}\]
C) \[\frac{56}{55}\]
D) \[\frac{55}{56}\]
Correct Answer: C
Solution :
[c] \[{{I}_{11}}=\int\limits_{0}^{1}{{{(1-{{x}^{5}})}^{11}}dx=\int\limits_{0}^{1}{(1-{{x}^{5}})\,{{(1-{{x}^{5}})}^{10}}}dx}\] \[\therefore \,\,\,\,\,\,\,{{I}_{11}}=\int\limits_{0}^{1}{{{(1-{{x}^{5}})}^{10}}dx-\int\limits_{0}^{1}{{{x}^{5}}{{(1-{{x}^{5}})}^{10}}dx}}\] ?(1) Now, \[\int{{{x}^{5}}{{(1-{{x}^{5}})}^{10}}dx=\int{x{{(1-{{x}^{5}})}^{10}}{{x}^{4}}dx}}\] \[=x\int{{{(1-{{x}^{5}})}^{10}}d\left( \frac{1-{{x}^{5}}}{-5} \right)}-\int{\left[ 1.{{(1-{{x}^{5}})}^{10}}d\left( \frac{1-{{x}^{5}}}{-5} \right) \right]}dx\]\[=-\frac{x}{55}{{(1-{{x}^{5}})}^{11}}+\frac{1}{55}\int{{{(1-{{x}^{5}})}^{11}}dx+c}\] From (1), we get \[{{I}_{11}}={{I}_{10}}+\left. \frac{x{{(1-{{x}^{5}})}^{11}}}{55} \right|_{0}^{1}-\frac{1}{55}{{I}_{11}}\] \[\Rightarrow \,\,\,\,\frac{56}{55}{{I}_{11}}={{I}_{10}}\]
You need to login to perform this action.
You will be redirected in
3 sec
