Title: Trigonometric Integrals

∫ (sin x)^m (cos x)ⁿ  dx

If m is an odd integer :

∫ (sin x)^m (cos x)ⁿ dx = ∫ (sin x)^m-1 (cos x)ⁿ sin  x dx

Change sin x to cos x (sin2 x = 1 - cos2 x)

take u = cos x , du = - sin x dx

If n is an odd integer

∫ (sin x)^m (cos x)ⁿ dx = ∫ (sin x)^m (cos x)^n-1 cos x dx

take u = sin x , du = cos x dx

Otherwise

use half-angle formulas for sin 2x and cos 2x to   reduce the exponents by one-half.

                           

Example 1:  m is an odd integer

∫ (sin x)³ (cos x)² dx

Let u = cos x,      du =  –sin x dx

∫ (sin x)² (cos x)² sin x dx = ∫ (1 – u²) u² (–du)           

       

Example 2: m and n are even

Example 3: m is an odd integer

∫ (tan x)^m (sec x)ⁿ dx = ∫ (tan x)^m-1 (sec x)^n-1 sec x tan x dx

take u = sec x , du = sec x tan x dx

Example 4: n is an even integer

∫ (tan x)^m (sec x)ⁿ dx = ∫ (tan x)^m (sec x)^n-2 (sec x)² dx

take u = tan x , du = (sec x)² dx

Example 5:

m is even and n is odd

There is no standard method of evaluation

Example 6:

∫ (tan x)³ (sec x)⁵ dx = ∫ (tan x)² (sec x)⁴ sec x tan x dx

 

Example  

∫ tan2 x sec4 x dx = ∫ tan2 x sec2 x sec2 x dx

Trigonometric Substitution :

i)   a² – x² let x = a sin θ

ii)  a² + x²        let x = a tan θ

iii) x² – a²        let x = a sec θ

Example: 

 

Let x = 2 sin θ, dx = 2 cos θ dθ

Plug them back to the integral :

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