Calculus: Early Transcendentals, Chapter 7, 7.2, Section 7.2, Problem 21

int tan(x) sec^3(x)dx
To solve, apply u-substitution method.
Let the u be:
u = sec(x)
Then, differentiate it.
du = tan(x) sec(x) dx
To be able to plug-in this to the integral, re-write the integrand as:
int tan(x) sec^3(x) dx = sec^2(x) * tan(x) sec(x) dx
Then, express the integrand in terms of u. So it becomes:
=int u^2 du
To take the integral of this, apply the formula int u^n du = u^(n+1)/(n+1)+C .
= u^3/3 + C
And, substitute back u= sec(x).
= (sec^3(x))/3+C
Therefore, int tan(x) sec^3(x) dx = (sec^3(x))/3+C .