How do you integrate #(tanx)^4 #?

1 Answer
Truong-Son N.
Oct 21, 2016

The trick with this one is to split it up into two #tan^2x# terms and use some identities.

#int tan^4xdx#

#= int tan^2xtan^2xdx#

#= int tan^2x(sec^2x - 1)dx#

#= int sec^2x(tanx)^2 - tan^2xdx#

#= int (tanx)^2sec^2x - (sec^2x - 1)dx#

Now for the first half, you can use u-substitution (let #u = tanx#, #du = sec^2xdx#), and for the second half, #intsec^2x = tanx#. Thus:

#=> int u^2du - int sec^2xdx + int1dx#

#= u^3/3 - tanx + x#

#= color(blue)(tan^3x/3 - tanx + x + C)#

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