Question

Integrate with respect to $x$:
${(e^x+\frac{1}{e^x})}^2$

Answer 1

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

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

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

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