Integration by Trigonometric Method (sine cosine)

Use trigonometric method when the integrand, f (x)  is a product of sin x and cos x or tan x and sec x. In this entry, we will discuss about sine cosine. Tangent secant will be discuss in the next entry.

How to evaluate  ∫ sinm x cosn x dx ? 

1.) Power of sin is odd : m = odd no. (and n = even no.)

• Split off a factor of sin x
• Use identity, substitute sin2 x = 1 - cos2 x
• Let u = cos x

eeeeeeExamples:  ∫ sin5 x cos2 x dx,  ∫ sin5 x dx,  
                             ∫ sin3 x cos2 x dx 

2.) Power of cos is odd : n = odd no. (and m = even no.)

• Split off a factor of sin x
• Use identity, substitute sin2 x = 1 - cos2 x
• Let u = cos x

eeeeeeExamples: ∫ sin6 x cos3 x dx, ∫ cos3 x dx,  
                            ∫ sin2 x cos5 x dx

3.) Both powers are odd

• Use either method (1) or (2) but it is easier to convert the term with the smallest power.

4.) Both powers are even

• Use half-angle identities

eeeeeeExamples: ∫ sin2 x cos2 x dx, ∫ sin4 x dx,  ∫ sin2 x dx

Click here for examples.