Question

The value(s) of $\int \frac{x^5}{x^2+1}dx$ is/are

• A. A trigonometric orrinverse trigonometric function
• B. $\frac{x^4}{4}-\frac{x^2}{2}+\frac{1}{2}\ln (x^2+1)+k$
• C. $\frac{{\sin }^{3}x}{3}-\frac{{\sin }^{5}x}{5}+k$
• D. $e^x\ln (x+2)+k$

Answer 1

Answer: (A), (B)

$\frac{x^4}{4}-\frac{x^2}{2}+\frac{1}{2}\ln (x^2+1)+k$

$I=\frac{1}{2}\int \frac{x^2.x^2.2x}{x^2+1}dx$
$=\frac{1}{2}\int \frac{(z-1{)}^2}{z}dz=\frac{1}{2}\int \frac{z^2+1-2z}{z}dz$
$=\frac{1}{2}[\ln z+\frac{z^2}{2}-2z]+k$
$=\frac{1}{2}\ln (z^2+1)+\frac{(x^2+1{)}^2}{4}-x^2-1+k$