$\int \frac{5\left(x^6+1\right)}{x^2+1} \mathrm{~d} x=$ (Where $C$ is a constant of integration.)

$\int \frac{5\left(x^6+1\right)}{x^2+1} \mathrm{~d} x=$ (Where $C$ is a constant of integration.)
  1. $5\left(x^7+1\right)+\log \left(x^2+1\right)+C$
  2. $x^5-\frac{5 x^3}{3}+5 x+C$
  3. $\frac{5 x^7}{7}+5 x+5 \tan ^{-1} x+C$
  4. $5 \tan ^{-1} x+\log \left(x^2+1\right)+C$

Solution

$\begin{aligned} & \int \frac{5\left(x^6+1\right)}{x^2+1} \mathrm{~d} x=\int \frac{5\left(x^2+1\right)\left(x^4-x^2+1\right)}{x^2+1} \mathrm{~d} x \\ & =5 \int\left(x^4-x^2+1\right) \mathrm{d} x \\ & =5\left\{\frac{x^5}{5}-\frac{x^3}{3}+x\right\}+C \\ & =x^5-\frac{5}{3} x^3+5 x+C\end{aligned}$

Asked in: MHT CET 2022 (07 Aug Shift 1)

Practice more Indefinite Integration questions on Aicharya