A) \[\frac{{{x}^{3}}}{3}-\frac{{{x}^{2}}}{2}+c\]
B) \[\frac{{{x}^{3}}}{3}-\frac{{{x}^{2}}}{2}+x-\log (x+1)+c\]
C) \[\frac{{{x}^{3}}}{3}-\frac{{{x}^{2}}}{2}+x+\log (x+1)+c\]
D) \[\frac{{{x}^{4}}}{2{{(x+1)}^{2}}}+c\]
E) \[\frac{{{x}^{3}}}{3}+\frac{{{x}^{4}}}{4}+c\]
Correct Answer: B
Solution :
Let \[I=\int{\frac{{{x}^{3}}}{x+1}}dx=\int{\frac{{{x}^{3}}+1-1}{x+1}}dx\] \[=\int{({{x}^{2}}+1-x)dx}-\int{\frac{1}{x+1}}dx\] \[=\frac{{{x}^{3}}}{3}-\frac{{{x}^{2}}}{2}+x-\log (x+1)+c\]
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