Indefinite integral using u-substitution with u = tan(x)

Let $u = \tan(x)$, so $du = \sec^2(x)\,dx$. Substituting into the integral, $\sec^2(x)\,dx$ becomes $du$ and $\tan^4(x)$ becomes $u^4$, so the integral transforms into $\int u^4\,du$.

Integrating with the power rule gives $\int u^4\,du = \dfrac{u^5}{5} + C$. Substituting back $u = \tan(x)$ yields the final answer $\dfrac{\tan^5(x)}{5} + C$.