Can we calculate integral of cos^4x as (cos^2x)^2 instead of cos^3x*cosx?

1 Answer
Jun 13, 2018

cos4xdx=3x8+sin(2x)4+sin(4x)32+C

Explanation:

Use the trigonometric identity:

cos2α=1+cos(2α)2

Then:

cos4xdx=(cos2x)2dx

cos4xdx=(1+cos(2x))24dx

using the linearity of the integral:

cos4xdx=14dx+12cos(2x)dx+14cos2(2x)dx

cos4xdx=x4+sin(2x)4+141+cos(4x)2dx

cos4xdx=x4+sin(2x)4+18dx+18cos(4x)dx

cos4xdx=3x8+sin(2x)4+sin(4x)32+C