What is the Integral of # (tan(x))^4#?

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Answer 1 · 2017-01-13T19:36:35

#tan^3x-tanx+x+C#.

Let #I=int(tanx)^4dx=inttan^4xdx#.

#:. I=inttan^2xtan^2xdx#

#=inttan^2x(sec^2x-1)dx#

#=inttan^2xsec^2xdx-inttan^2xdx#

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

#=J-intsec^2xdx+int1dx#

#=J-tanx+x#, where,

#J=inttan^2xsec^2xdx#

To find #J," we subst. "y=tanx," so that, "dy=sec^2xdx#.

#:. J=inty^2dy=y^3/3=1/3tan^3x#. Finally, we have,

#I=1/3tan^3x-tanx+x+C#.

Enjoy Maths.!