Question

Evaluate the ∫e^t dt/cuberoot (1-e^t)

Answer 1

Solution

Substitution x = e^t :

$\int \frac{e^t dt}{\sqrt[3]{1-e^t}}=\int\frac{dx}{\sqrt[3]{1-x}}=\int(1-x)^{-1/3}dx= -\frac{3}{2}(1-x)^{2/3}+C=-\frac{3}{2}(1-e^t)^{2/3}+C$

Answer

$\int \frac{e^t dt}{\sqrt[3]{1-e^t}}=-\frac{3}{2}(1-e^t)^{2/3}+C$