KVPY Sample Paper KVPY Stream-SX Model Paper-11

If \[f(x)=\int{\frac{5{{x}^{8}}+7{{x}^{6}}}{{{({{x}^{2}}+1+2{{x}^{7}})}^{2}}}dx,}\]\[(x\ge 0)\] and \[f(0)=0.\] then the value of\[f(1)\] is:

A) \[-\frac{1}{2}\]

B) \[\frac{1}{2}\]

C) \[-\frac{1}{4}\]

D) \[\frac{1}{4}\]

Correct Answer: D

Solution :

\[\int{\frac{5{{x}^{8}}+7{{x}^{6}}}{{{({{x}^{2}}+1+2{{x}^{7}})}^{2}}}dx}=\int{\frac{5{{x}^{-\,6}}+7{{x}^{-\,8}}}{{{\left( \frac{1}{{{x}^{7}}}+\frac{1}{{{x}^{5}}}+2 \right)}^{2}}}dx}\]

\[=\,\frac{1}{2+\frac{1}{{{x}^{5}}}+\frac{1}{{{x}^{7}}}}+C\]

As \[f(0)-0,\] \[f(x)=\frac{{{x}^{7}}}{2{{x}^{7}}+{{x}^{2}}+1}\]

\[f(1)=\frac{1}{4}.\]

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KVPY Sample Paper KVPY Stream-SX Model Paper-11

question_answer If \[f(x)=\int{\frac{5{{x}^{8}}+7{{x}^{6}}}{{{({{x}^{2}}+1+2{{x}^{7}})}^{2}}}dx,}\]\[(x\ge 0)\] and \[f(0)=0.\] then the value of\[f(1)\] is: A) \[-\frac{1}{2}\] B) \[\frac{1}{2}\] C) \[-\frac{1}{4}\] D) \[\frac{1}{4}\]

If \[f(x)=\int{\frac{5{{x}^{8}}+7{{x}^{6}}}{{{({{x}^{2}}+1+2{{x}^{7}})}^{2}}}dx,}\]\[(x\ge 0)\] and \[f(0)=0.\] then the value of\[f(1)\] is:

A) \[-\frac{1}{2}\]

B) \[\frac{1}{2}\]

C) \[-\frac{1}{4}\]

D) \[\frac{1}{4}\]