How do you integrate int cos^2(x) tan^3(x) dx?

1 Answer
Jul 23, 2016

I got:

cos^2x/2 - ln|cosx| + C

and Wolfram Alpha agrees.


Note that tan^3x = sin^3x/cos^3x. Therefore:

color(blue)(int cos^2xtan^3xdx)

= int (sin^3x)/cosxdx

Then, you can use u-substitution. When u = cosx, du = -sinxdx. To do that, you need to turn sin^2x into 1-cos^2x.

= int ((1-cos^2x)sinx)/(cosx)dx

Hence:

= -int ((1-cos^2x)(-sinx))/(cosx)dx

= -int (1-u^2)/(u)du

= int (u^2 - 1)/(u)du

= int u - 1/udu

= u^2/2 - ln|u|

Re-substitute to get:

=> color(blue)(cos^2x/2 - ln|cosx| + C)