To integrate (tan(x))^2(sec(x)), start by rewriting tan(x) and sec(x) in terms of sine and cosine, leading to the expression sin^2(x)/cos^3(x). By factoring out a cosine, the integral simplifies to sin^2(x)cos(x)/cos^4(x), which can be transformed using the substitution u = sin(x). Alternatively, using the substitution u = tan(x/2) allows for a different approach, yielding an integral that can be solved through partial fractions. The final result involves integrating sec^3(x) and sec(x), with the latter evaluated using logarithmic identities.