How do you integrate # tan^3x#?

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Answer 1 · 2017-02-12T15:06:13

#1/2tan^2x+ln|secx|+C#

by the use of trig identities alternative equivalent solns can be found

#inttan^3xdx#

now#" "1+tan^2x=sec^2x=tan^2x=sec^2x-1#

#inttan^3xdx=inttan^2xtanxdx#

#=int(sec^2x-1)tanxdx#

#=int(sec^2xtanx)dx-inttanxdx#

now

#d/(dx)(tan^2x)=2tanxsec^2x#

#:.=int(sec^2xtanx)dx=1/2tan^2x#

#inttanxdx=intsinx/cosxdx=-ln|cosx|=ln|secx|#

so the complete soln

#inttan^3xdx#

#=1/2tan^2x+ln|secx|+C#