Question

How do I evaluate the indefinite integral `int(tan^2(x)+tan^4(x))^2dx` ?

Answer 1

`\int(\tan^2(x)+\tan^4(x))^2 dx=\int((tan^2 x(1+tan^2 x))^2 dx`
`=\int tan^4 x(1+tan^2 x) (1+tan^2x) dx`
`=\int(tan^4 x+tan^6 x) (1+tan^2 x) dx`
Now, note that `(tan x)'=sec^2 x=1+tan^2 x` (see derivatives of trig functions ).

Thus, the substitution `u=tan x` yields `du=(1+tan^2 x) dx` and
`\int(\tan^2(x)+\tan^4(x))^2 dx=\int u^4+u^6 du=u^5/5+u^7/7+C`
`=(\tan^5 x)/5+(\tan^7 x)/7+C`
using the power rule for integration (see, e.g, antiderivatives )