Powers of Trig Functions Powers of Sin and Cos Example:
Evaluate
sin5x dx
Exercise
Example
Solution = (1/2 - 1/2 cos(2x))2 dx
=
[1/4 - 1/2 cos(2x) + 1/4 cos2(2x)] dx The integral becomes
1/4 x - 1/4 sin(2x) + 1/4
[1/2 + 1/2 cos(4x)] dx = 1/4 x - 1/4 sin(2x) + 1/8 x + 1/32 sin(4x) + C
We see that if the power is odd we can pull out one of the sin functions
and convert the other to an expression involving the cos function only.
Then use
If the power is even, we must use the trig identities
Mixed Powers of Sin and Cos
sin2x cos3x dx sin2x (1 - sin2 x) cosx dx Let u = sin x, du = cos x dx We have
u2(1 - u2)du
= 1/3 sin3x -
1/5 sin5x + C
Exercise Powers of Tangents and Secants To integrate powers of tangents and secants we use the formula
tan2 x
= sec2 x - 1
Solution
= tan2 x sec2 x dx
- tan2 x dx
For the first integral let
We have = 1/3 u3 - tanx + x + C
= 1/3 tan3 x - tan x
+ x + C
Exercise
Mixed Angles
We have the following formulas:
sin(3x)sin(4x)dx
We integrate the first by letting u = -x and the second by letting
u = 7x to get
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