Calculus: Early Transcendentals, Chapter 7, 7.2, Section 7.2, Problem 24

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 .