Agora aplicamos a fórmula de integração de \operatorname{sec}(u):
Passo 4
\frac{1}{5}\,\int \operatorname{sec}(u)\,du = \frac{1}{5}\,\ln\bigl|\operatorname{sec}(u)+\tan(u)\bigr| + C.
Voltando à variável x:
Passo 5
\frac{1}{5}\,\ln\bigl|\operatorname{sec}\bigl(5x-\pi\bigr)+\tan\bigl(5x-\pi\bigr)\bigr| + C.
Como \operatorname{sec}(\theta-\pi)=-\operatorname{sec}\theta e \tan(\theta-\pi)=\tan\theta, dentro do valor absoluto o sinal “–” desaparece, ficando equivalente a
\frac{1}{5}\,\ln\bigl|\operatorname{sec}(5x)+\tan(5x)\bigr| + C.
Resposta
\int \operatorname{sec}\bigl(5x-\pi\bigr)\,dx = \frac{1}{5}\,\ln\bigl|\operatorname{sec}\bigl(5x-\pi\bigr)+\tan\bigl(5x-\pi\bigr)\bigr| + C
que pode ser simplificado para
\frac{1}{5}\,\ln\bigl|\operatorname{sec}(5x)+\tan(5x)\bigr| + C.