Question

Evaluate
$\int e^x(\tan x+1)secxdx$

Answer 1

$\begin{array}{c}I=\int e^x\left(\tan x+1\right)\sec xdx\\ =\int e^x\tan x\sec xdx+\int e^x\sec xdx\end{array}$

According to the integration by parts,

$\begin{array}{c}\int uvdx=u\int vdx-\int \left(\frac{du}{dx}\times \int vdx\right)dx\\ Here,\\ u=\sec x\\ v=e^x\\ So,\\ \int \sec x{e}^{x}dx=\sec x\int e^{x}dx-\int \left(\frac{d\sec x}{dx}\times \int e^{x}dx\right)dx\\ =e^x\sec x-\int \left(\sec x\tan x{e}^x\right)dx\\ So,\\ I=\int \left(\sec x\tan x{e}^x\right)dx+e^x\sec x-\int \left(\sec x\tan x{e}^x\right)dx\\ =e^x\sec x\end{array}$