Question

Integrate the following function
$\frac{\cos x}{\cos (x+a)}$

Answer 1

$\int \frac{cosx}{\cos (x+a)}dx$

Substitute, $x+a=t$

$dt=dx$

$\therefore$ $\int \frac{\cos \left(x\right)}{\cos \left(t\right)}dt$

$=\int \frac{\cos (t-a)}{cost}dt$

$=\int \left[\frac{cost.cosa+sint.sina}{cost}\right]dt$

$=\int [cosa+tantsina]dt$

$=cosa\int dt+sina\int tantdt$

$=cosat+sinaInsect+C$

$=tcosa+sina.Insect+C$

$=(x+a)cosa+sinaIn\sec (x+a)+C$