Question

$\int e^x{\left(\frac{x-1}{x^2+1}\right)}^{2}dx$

• A. $\frac{{xe}^x}{x^2+1}+c$
• B. $\frac{x^2{e}^x}{x^2+1}+c$
• C. $\frac{e^x}{x^2+1}+c$
• D. $\frac{e^x(x-3)}{x+1}+c$

Answer 1

The correct option is C $\frac{e^x}{x^2+1}+c$

$\int {e^x\left(\frac{x-1}{x^2+1}\right)}^{2}dx$

$=\int e^x\frac{{\left(x-1\right)}^2}{{\left(x^2+1\right)}^2}dx$

$=\int e^2\left[\frac{\left(x^2+1\right)-2x}{{\left(x^2+1\right)}^2}\right]$

$=\int e^x\left[\frac{x^2+1}{{\left(x^2+1\right)}^2}-\frac{2x}{{\left(x^2+1\right)}^2}\right]$

$=\int e^x\left[\frac{1}{\left(x^2+1\right)}-\frac{2x}{{\left(x^2+1\right)}^2}\right]$

$=\frac{e^x}{x^2+1}+c$