Answer to Question #162022 in Calculus for Phyroe

Question #162022

Evaluate the ∫e^t dt/cuberoot (1-e^t)


1
Expert's answer
2021-02-24T12:27:11-0500

Solution

Substitution x = et :

etdt1et3=dx1x3=(1x)1/3dx=32(1x)2/3+C=32(1et)2/3+C\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

etdt1et3=32(1et)2/3+C\int \frac{e^t dt}{\sqrt[3]{1-e^t}}=-\frac{3}{2}(1-e^t)^{2/3}+C


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