Powers of Trig Functions

Powers of Sin and Cos

Example:

Evaluate

sin5x dx

we write

sin5x dx

=  sin4x sinx dx

=  (sin2 x)2sinx dx

(1-cos2x)2 sinx dx        u = cosx    du = -sinx dx

=  -(1 - u2)2 du

=  -[1 - 2u2 + u4] du

=  -u + 2/3 u3 - 1/5 u5 + C

=  -cosx + 2/3 cos3x - 1/5 cos5x + C

Exercise

Evaluate

cos7x dx

Example

Evaluate

sin4x dx

Solution

We write

sin4x dx  =  (sin2x)2 dx

=  (1/2 - 1/2 cos(2x))2 dx

= [1/4 - 1/2 cos(2x) + 1/4 cos2(2x)] dx

For the second integral let

u = 2x     dx = 1/2du

The integral becomes

1/4 x - 1/4 sin(2x) + 1/4 [1/2 + 1/2 cos(4x)] dx

Let u = 4x dx = 1/4du

= 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

u = cos x.

If the power is even, we must use the trig identities

 sin2x  =  1/2 - 1/2 cos(2x)  and cos2x  =  1/2 + 1/2 cos(2x)

This method will always work and is always long and tedious.

Mixed  Powers of Sin and Cos

Example

Evaluate

sin2x cos3x dx

We pull out a cos x and convert the cos2 x to 1 - sin2 x:

sin2x (1 - sin2 x) cosx dx    Let u = sin x,   du = cos x dx

We have

u2(1 - u2)du

=  u2 - u4 du = 1/3 u3  - 1/5 u5 + C

=  1/3 sin3x - 1/5 sin5x + C

Exercise

sin5x cos2x dx

Powers of Tangents and Secants

To integrate powers of tangents and secants we use the formula

tan2 x  =  sec2 x - 1

Example

Evaluate

tan4 x dx

Solution

We write

tan2 x tan2 x dx

=  tan2 x (sec2 x - 1)

= tan2 x sec2 x dx  -  tan2 x dx

=  tan2 x sec2 x dx  -  (sec2 x - 1) dx

For the first integral let

u = tan x du = sec2 x dx.

We have

u2 du - tanx + x

= 1/3 u3 - tanx + x + C

= 1/3 tan3 x - tan x + x + C

Exercise

sec5 x tan3 x dx.

Mixed Angles

We have the following formulas:

 sin(mx) sin(nx) = 1/2[cos[(m - n)x] - cos[(m + n)x]] sin(mx) cos(nx) = 1/2[sin[(m - n)x] + sin[(m + n)x]] cos(mx) cos(nx) = 1/2[cos[(m - n)x] + cos[(m + n)x]]

Example

sin(3x)sin(4x)dx

= 1/2 [cos(-x) - cos(7x)] dx

We integrate the first by letting u = -x and the second by letting u = 7x to get

1/2 -sin(-x) + 1/7 sin(7x) + C

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