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

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Answer 1 · 2016-10-21T19:55:03

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:

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

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

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

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