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  • integrate x cube by x plus 1 equals

\int \frac{x^{3}}{x+1} is equal to

\\A. x+\frac{x^{2}}{2}+\frac{x^{3}}{3}-\log |1-x|+C\\ B. x+\frac{x^{2}}{2}-\frac{x^{3}}{3}-\log |1-x|+C\\ C. x-\frac{x^{2}}{2}-\frac{x^{3}}{3}-\log |1+x|+C\\ D. x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\log |1+x|+C\\

Answers (1)

D)

Given:  \int \frac{x^{3}}{x+1}

\\ \frac{x^{3}}{x+1}=\frac{x^{3}+1-1}{x+1} \\ \Rightarrow \frac{x^{2}+1}{x+1}-\frac{1}{x+1}=\frac{(x+1)\left(x^{2}-x+1\right)}{x+1}-\frac{1}{x+1}

\\ \int \frac{x^{3}}{x+1} d x=\int\left(\left(x^{2}-x+1\right)-\frac{1}{x+1}\right) d x \\ \Rightarrow \int\left(x^{2}-x+1\right) d x-\int \frac{1}{x+1} d x=\frac{x^{3}}{3}-\frac{x^{2}}{2}+x-\ln |1+x|+c

 

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