Question

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

Answer 1

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

Explanation:

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.!