Integration of trigonometric functions

[Indir]

TL;DR

Integration problem

Was solving a problem in mathematics and came across the following integration. Unable to move further. Can somebody provide answer for the following ( a and b are constants ).

 

[anuttarasammyak]

Why don not you try substitution
a-b \cos x = u?

 

[fresh_42]

A good plan to tackle such questions is: remove what disturbs the most! That often helps to get into the problem. If you have trig functions then it is always good to keep the Weierstraß substitution in mind; not here but in general.

 

[WWGD]

On its own, just as a trick, $sinxcosx=\frac{sin2x}{2}$, with simple integral $\frac{-Cos2x}{2}$
But, yes, that denominator kills it. Maybe Fresh can write an insight on integrating expressions a/b from the respective integrals of a,b , right, Fresh? ;)

 

[fresh_42]

On its own, just as a trick, $sinxcosx=\frac{sin2x}{2}$, with simple integral $\frac{-Cos2x}{2}$
But, yes, that denominator kills it. Maybe Fresh can write an insight on integrating expressions a/b from the respective integrals of a,b , right, Fresh? ;)

The difficulty with integrating products (and likewise quotients) arises from the fact that differentiation is a derivation. The Jacobi identity / Leibniz rule / product rule rules this world and not the chain rule.
$$
D(f\cdot g) = Df \cdot g + f\cdot Dg
$$
We can sometimes use the fact the $D\sin= \cos$ and $D\cos= -\sin$ and in the case of trigonometric functions. Here is an example:
https://www.physicsforums.com/insig...tion/#Integration-by-Parts-–-The-Leibniz-Rule