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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