`int (tan^2(x) + tan^4(x)) dx` Evaluate the integral

`int (tan^2(x)+tan^4(x))dx`

Factor out the GCF of the two terms.

`= int tan^2(x)(1 + tan^2(x)) dx`

Then, apply the trigonometric identity `sec ^2 theta = tan^2 theta + 1` .

`= int tan^2(x) sec^2(x)dx`

To take the integral of this, use u-substitution method. So let u be:

     `u= tan(x)`

Then, differentiate u.

     `du = sec^2(x) dx`

Plugging them, the integral becomes:

`= int u^2du`

Then, apply the formula `int u^n du = u^(n+1)/(n+1)+C` .

`= u^3/3 + C`

And, substitute back u = tan(x).

`= (tan^3(x))/3 + C`

Therefore, `int (tan^2(x)+tan^4(x))dx = (tan^3(x))/3+C` .