What is the integral of (1-x)/(3e^x) ?

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

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integral of (1-x)/(3e^x)

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

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

Solution steps

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

Apply u-substitution

= \frac{1}{3} \cdot \int \frac{1 - ln ( u )}{u ^{2}} d u

Apply Integration By Parts

= \frac{1}{3} (- \frac{1 - ln ( u )}{u} - \int \frac{1}{u ^{2}} d u)

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

= \frac{1}{3} (- \frac{1 - ln ( u )}{u} - (- \frac{1}{u}))

Substitute back

u = e^{x}

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

Simplify

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

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

Add a constant to the solution

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

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