Powers of Trig Functions

  1. Powers of Sin and Cos

    Example:

    Evaluate int sin5x dx we write

    Exercise:  int cos7x dx

    Example:  Evaluate int sin4x dx

    Solution:  We write  int sin4x dx = int (sin2x)2 dx

    =  int (1/2 - 1/2 cos(2x))2 dx

    = int 1/4 - 1/2 cos(2x) + 1/4 cos2(2x) dx  For the second integral let u = 2x dx = 1/2du

    = 1/4 x - 1/4 sin(2x) + 1/4 int  1/2 + 1/2 cos(4x)  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.

  2. Mixed  powers of Sin and Cos

    Example:  Evaluate

    int sin2x cos3x dx.  We pull out a cos x and convert the cos2 x to 1 - sin2 x:

    int sin2x (1 - sin2 x) cosx dx    Let u = cosx,   du = -sinx dx

    We have

    int u2(1 - u2)du = int u2 - u4 du = 1/3 u3  - 1/5 u5 + C

    =  1/3 cos3x - 1/5 cos5x + C     

    Exercise:  int sin5x cos2x dx

  3. Powers of Tangents and Secants   

    To integrate powers of tangents and secants we use the formula

    tan2 x = sec2 x - 1

    Example:   Evaluate int tan4 x dx

    Solution:  We write int tan2 x tan2 x dx = int tan2 x (sec2 x - 1)

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

    For the first integral let u = tan x du = sec2 x dx.

    We have int u2 du - tanx - x

    = 1/3 u3 - tanx - x + C

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

    Exercise:  int sec5 x tan3 x dx.

  4. 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:

    int sin(3x)sin(4x)dx = 1/2 int cos(-x) - cos(7x) dx

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