Transcript 6.2trigonometric

6.2 Trigonometric Integrals
How to integrate
powers of sinx and cosx
(i) If the power of cos x is odd, save one cosine factor and use cos2x
= 1 - sin2x to express the remaining factors in terms of sin x. Then
substitute u = sin x.
(ii) If the power of sin x is odd, save one sine factor and use sin2x = 1
- cos2x to express the remaining factors in terms of cos x. Then
substitute u = cos x.
(iii) If the powers of both sine and cosine are even, use the half-angle
identities:
sin2x = 0.5(1 – cos 2x)
cos2x = 0.5(1 + cos 2x)
It is sometimes helpful to use the identity:
sin x cos x = 0.5 sin 2x
Example: Evaluate the integral
(the solution on the board)
 sin
6
3
x cos x dx
How to integrate
powers of tanx and secx
(i) If the power of sec x is even,
save a factor of sec2x
and use sec2x = 1 + tan2x
to express the remaining factors in terms of tan x.
Then substitute u = tan x.
(ii) If the power of tan x is odd,
save a factor of sec x tan x
and use tan2x = sec2x – 1
to express the remaining factors in terms of sec x.
Then substitute u = sec x.
Example: Evaluate the integral
(the solution on the board)
3
tan
 x sec x dx