What is the integral of 3/(\sqrt[3]{e^x)} ?

The integral of 3/(\sqrt[3]{e^x)} is 3(x/(e^{x/3)}-e^{-x/3}(x+3))+C

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integral of 3/(\sqrt[3]{e^x)}

\int \frac{3}{3 e ^{x}} d x

3 (\frac{x}{e ^{\frac{x}{3}}} - e^{- \frac{x}{3}} (x + 3)) + C

Solution steps

\int \frac{3}{3 e ^{x}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int \frac{1}{3 e ^{x}} d x

Apply Integration By Parts

: \frac{x}{e ^{\frac{x}{3}}} - \int - \frac{x}{3 e ^{\frac{x}{3}}} d x

= 3 (\frac{x}{e ^{\frac{x}{3}}} - \int - \frac{x}{3 e ^{\frac{x}{3}}} d x)

\int - \frac{x}{3 e ^{\frac{x}{3}}} d x = e^{- \frac{x}{3}} (x + 3)

= 3 (\frac{x}{e ^{\frac{x}{3}}} - e^{- \frac{x}{3}} (x + 3))

Add a constant to the solution

= 3 (\frac{x}{e ^{\frac{x}{3}}} - e^{- \frac{x}{3}} (x + 3)) + C

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What is the integral of 3/(\sqrt[3]{e^x)} ?
The integral of 3/(\sqrt[3]{e^x)} is 3(x/(e^{x/3)}-e^{-x/3}(x+3))+C