Question

Integrate cosx/cos(x+a)dx

Answer 1

Substitute t for (x+a). Hope this is helpful.

Answer 2

Answer-

$\int\ \frac{\cos x}{\cos (x+a)} \, dx=(x+a)\cos a+\sin a.\log \sec (x+a)+c$

Solution-

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

Putting (x+a) as t,

$\Rightarrow x+a =t\ \ \ \ \ \ \ \ \ \ \Rightarrow x=t-a \\ \Rightarrow dx =dt$

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

$=\int\ \frac{\cos t.\cos a+\sin t.\sin a}{\cos t} \, dt$

$=\int\ (\frac{\cos t.\cos a}{\cos t} +\frac{\sin t.\sin a}{\cos t}) \, dt$

$=\int\ \cos a\, dt+\int \tan t.\sin a \, dt$

$=\cos a\int dt+\sin a\int \tan t \, dt$

$=t\cos a+\sin a.\log \sec t+c$

$=(x+a)\cos a+\sin a.\log \sec (x+a)+c$