KVPY Sample Paper KVPY Stream-SX Model Paper-6

\[\int{\frac{{{x}^{4}}+4}{{{x}^{2}}-2x+2}\,\,dx}\] is

A) \[\frac{{{x}^{3}}}{2}+{{x}^{2}}+2x+c\]

B) \[\frac{{{x}^{3}}}{3}+{{x}^{2}}+2x+c\]

C) \[\frac{{{x}^{3}}}{3}+\frac{{{x}^{2}}}{2}+x+c\]

D) \[\frac{{{x}^{3}}}{3}+{{x}^{2}}-2x+c\]

Correct Answer: B

Solution :

\[{{x}^{4}}+4={{x}^{4}}+4+4{{x}^{2}}-4{{x}^{2}}\,{{({{x}^{2}}+2)}^{2}}-{{(2x)}^{2}}\]\[=({{x}^{2}}-2x+2)\]

\[\therefore \] \[I=\int{({{x}^{2}}+2x+2)\,dx=\frac{{{x}^{3}}}{3}+{{x}^{2}}+2x+c}\]