Question

Resolver a integral usando o método de substituição:
Integral de dt/(3t-5)^3

Answer 1

Calcular a integral indefinida:

     $\displaystyle\int\frac{1}{(3t-5)^3}\,dt\\\\\\ =\int(3t-5)^{-3}\,dt\\\\\\ =\int\frac{1}{3}\cdot 3(3t-5)^{-3}\,dt\\\\\\ =\frac{1}{3}\int (3t-5)^{-3}\cdot 3\,dt$



Faça a seguinte substituição:

     $3t-5=u\quad\Rightarrow\quad 3\,dt=du$



e a integral fica

     $\displaystyle=\frac{1}{3}\int u^{-3}\,du\\\\\\ =\frac{1}{3}\cdot \frac{u^{-3+1}}{-3+1}+C\\\\\\ =\frac{1}{3}\cdot \frac{u^{-2}}{-2}+C\\\\\\ =-\,\frac{1}{6}\,u^{-2}+C\\\\\\ =-\,\frac{1}{6u^2}+C$

     $=-\,\dfrac{1}{6(3t-5)^2}+C\quad\longleftarrow\quad\textsf{resposta.}$



Bons estudos! :-)