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

The integral of x/(3e^x) is (-x-1)/(3e^x)+C

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integral of x/(3e^x)

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

\frac{- x - 1}{3 e ^{x}} + C

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

\frac{1}{3} (- \frac{x}{e ^{x}} - \frac{1}{e ^{x}}) : \frac{- x - 1}{3 e ^{x}}

= \frac{- x - 1}{3 e ^{x}}

Add a constant to the solution

= \frac{- x - 1}{3 e ^{x}} + C

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