show instructions\nquestion 9\nif ∫xsec²xdx = xtanx + f(x) + c, then f(x) =\no a ln|cosx|\no b -ln|cosx|\no…

Explanation:

Step1: Use integration - by - parts

The formula for integration by parts is $\int u dv=uv-\int v du$. Let $u = x$ and $dv=\sec^{2}x dx$. Then $du = dx$ and $v=\tan x$. So, $\int x\sec^{2}x dx=x\tan x-\int\tan x dx$.

Step2: Integrate $\tan x$

We know that $\tan x=\frac{\sin x}{\cos x}$. Let $u = \cos x$, then $du=-\sin x dx$. So, $\int\tan x dx=\int\frac{\sin x}{\cos x}dx=-\int\frac{du}{u}=-\ln|u|=-\ln|\cos x|+C$. Then $\int x\sec^{2}x dx=x\tan x+\ln|\cos x|+C$.

Answer:

A. $\ln|\cos x|$