Question

evaluate intregation of cosx /cos(x-a) ×dx​

Answer 1

To Integrate :

$\star \int \: \frac{ cosx}{ cos(x - a )} dx$

By substitution :

Let , x- a = t Or x = t+a

=> dx = dt

$\implies \: \int \: \frac{cos \: (t + a)}{cos \: t} dt$

As we know that :

$\star \boxed{ \bold{ \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) }}$

$\implies \: \int \: \frac{ \cos(t) \cos(a) - \sin(t) \: \sin(a) }{ \cos(t) } dt \\ \\ \implies \: \cos(a) \int \: dt - \sin(a) \int \frac{ \sin(t) }{ \cos(t) } dt \\ \\ \implies \: \cos( a) \int dt - \sin(a) \int \: \tan(t) dt \\ \\ \implies \: \cos(a) (t) - \sin(a) ( ln( \sec(t) ) ) + c$

• Now , put t = x-a we get ,

$\implies(x - a) \cos(a) - \sin(a) ( ln( \sec(x - a) ) ) + c$

Formula :

$\star \boxed{ \bold{ \int \: tan \: xdx = ln( sec \: x) + c}}$