Obter a primitiva de $f(x) = \left(\sec \dfrac{\pi x}{2}\right)\left(\tan \dfrac{\pi x}{2}\right)$.

$f(x) = \dfrac{\sin \dfrac{\pi x}{2}}{\cos^2 \dfrac{\pi x}{2}}$

Seja $u = \cos \dfrac{\pi x}{2}$, $du = -\dfrac{\pi}{2}\sin \dfrac{\pi x}{2}\ dx$.

$\displaystyle\int f(x)\ dx\ =\ -\dfrac{2}{\pi} \displaystyle\int \dfrac{du}{u^2}\ =\ \dfrac{2}{\pi u} + c = \fbox{$\dfrac{2\sec \dfrac{\pi x}{2}}{\pi} + c$}$