Calcule a integral.
\int \operatorname{sec} ^{3}(3 x) d x

Passo 1

O enunciado pede para calcular
\int \operatorname{sec}^{3}(3x)\,dx.
O conteúdo principal aqui é a técnica de substituição para lidar com o argumento 3x e depois a integral de \operatorname{sec}^{3}u usando integração por partes.

Passo 2

Fazemos a substituição
u = 3x, \qquad du = 3\,dx \quad\Longrightarrow\quad dx = \frac{du}{3}.
Assim a integral vira
\int \operatorname{sec}^{3}(3x)\,dx = \int \operatorname{sec}^{3}(u)\,\frac{du}{3} = \frac{1}{3}\int \operatorname{sec}^{3}(u)\,du.

Passo 3

Agora calculamos
I = \int \operatorname{sec}^{3}(u)\,du
por partes. Escrevemos
I = \int \operatorname{sec}(u)\,\operatorname{sec}^{2}(u)\,du,
e tomamos
\begin{cases} A = \operatorname{sec}(u), & dB = \operatorname{sec}^{2}(u)\,du,\\\ dA = \operatorname{sec}(u)\tan(u)\,du, & B = \tan(u). \end{cases}
Pela fórmula de partes,
I = A\,B - \int B\,dA = \operatorname{sec}(u)\tan(u) - \int \tan(u)\bigl[\operatorname{sec}(u)\tan(u)\bigr]\,du.
Como \tan^{2}(u)=\operatorname{sec}^{2}(u)-1, temos
\int \operatorname{sec}(u)\tan^{2}(u)\,du = \int \operatorname{sec}(u)\bigl[\operatorname{sec}^{2}(u)-1\bigr]\,du = \int \operatorname{sec}^{3}(u)\,du - \int \operatorname{sec}(u)\,du = I - \int \operatorname{sec}(u)\,du.
Logo,
I = \operatorname{sec}(u)\tan(u) - \bigl[I - \int \operatorname{sec}(u)\,du\bigr] = \operatorname{sec}(u)\tan(u) - I + \int \operatorname{sec}(u)\,du,
2I = \operatorname{sec}(u)\tan(u) + \int \operatorname{sec}(u)\,du = \operatorname{sec}(u)\tan(u) + \ln\bigl|\operatorname{sec}(u)+\tan(u)\bigr|,
pois \displaystyle\int \operatorname{sec}(u)\,du=\ln\bigl|\operatorname{sec}(u)+\tan(u)\bigr|. Portanto
I = \frac{1}{2}\Bigl(\operatorname{sec}(u)\tan(u) + \ln\bigl|\operatorname{sec}(u)+\tan(u)\bigr|\Bigr).

Passo 4

Voltando à variável x e lembrando do fator \tfrac{1}{3}, obtemos
\int \operatorname{sec}^{3}(3x)\,dx = \frac{1}{3}I = \frac{1}{3}\cdot\frac{1}{2}\Bigl(\operatorname{sec}(u)\tan(u) + \ln\bigl|\operatorname{sec}(u)+\tan(u)\bigr|\Bigr)+C.
Substituindo u=3x, fica
\int \operatorname{sec}^{3}(3x)\,dx = \frac{1}{6}\operatorname{sec}(3x)\tan(3x) + \frac{1}{6}\ln\bigl|\operatorname{sec}(3x)+\tan(3x)\bigr| + C.

Resposta

\boxed{ \int \operatorname{sec}^{3}(3x)\,dx = \frac{1}{6}\operatorname{sec}(3x)\tan(3x) + \frac{1}{6}\ln\bigl|\operatorname{sec}(3x)+\tan(3x)\bigr| + C. }