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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