Unit 8 - Objective 5 - Odd powers of Sine and/or Cosine



In the first section of this unit, you integrated powers of sine as long as there was a single power of cosine to be the du.



When the integral contains only an odd power of sine (or cosine) then use the trig identities:



Peel off and save one sine (or cosine) to be the "du" and replace the rest.



Problem 1:

sin³x dx = sin²x sin x dx save sin x

= (1 - cos²x) sin x dx multiply across

= (sin x - cos²x sin x) dx integrate one term at a time

= sin x dx - cos²x sin x dx

ok u = cos x, du = -sin x dx, "fix" the -1

= sin x dx - (-1) cos²x (-1) sin x dx

= -cos x - (-1) + C

= -cos x + (1/3) cos³x + C


Problem 2:

cos³(5t) dt = (cos²5t) cos 5t dt = (1 - sin²5t) cos 5t dt

= cos 5t dt - sin²5t cos 5t dt

=

= (1/5) sin 5t - (1/5) + C

= (1/5) sin 5t - (1/15) sin³5t + C




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