What is the integral of x/(3e^{2x)} ?

The integral of x/(3e^{2x)} is 1/3 (-1/2 e^{-2x}x-1/4 e^{-2x})+C

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integral of x/(3e^{2x)}

\int \frac{x}{3 e ^{2 x}} d x

\frac{1}{3} (- \frac{1}{2} e^{- 2 x} x - \frac{1}{4} e^{- 2 x}) + C

Solution steps

\int \frac{x}{3 e ^{2 x}} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{x}{e ^{2 x}} d x

Apply Integration By Parts

: - \frac{1}{2} e^{- 2 x} x - \int - \frac{1}{2} e^{- 2 x} d x

= \frac{1}{3} (- \frac{1}{2} e^{- 2 x} x - \int - \frac{1}{2} e^{- 2 x} d x)

\int - \frac{1}{2} e^{- 2 x} d x = \frac{1}{4} e^{- 2 x}

= \frac{1}{3} (- \frac{1}{2} e^{- 2 x} x - \frac{1}{4} e^{- 2 x})

Add a constant to the solution

= \frac{1}{3} (- \frac{1}{2} e^{- 2 x} x - \frac{1}{4} e^{- 2 x}) + C

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What is the integral of x/(3e^{2x)} ?
The integral of x/(3e^{2x)} is 1/3 (-1/2 e^{-2x}x-1/4 e^{-2x})+C