Question

Evaluate $\int \left(x\right(\tan x)+\ln (\sec x\left)\right)dx$
(where $C$ is constant of integration)

• A. $x\ln \left(\sec x\right)+C$
• B. $-x\ln \left(\sec x\right)+C$
• C. $\ln \left(\sec x\right)+C$
• D. $-\ln \left(\sec x\right)+C$

Answer 1

The correct option is A $x\ln \left(\sec x\right)+C$

Let $I=\int \left(x\right(\tan x)+\ln (\sec x\left)\right)dx$Clearly $I$ is of the form :

$\int \left(f\right(x)+x{f}^{\prime }\left(x\right))dx=xf(x)$
We know, if $f\left(x\right)=\ln \left(\sec x\right)$ then $f^{\prime }\left(x\right)=\tan x$
$\therefore I=x\ln \left(\sec x\right)+C$