Evaluate integral of xe^(x/3) with respect to x

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

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{\frac{x}{3}}$ .

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

Simplify.

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Combine $\frac{1}{3}$ and $x$ .

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

Move $3$ to the left of $x$ .

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

Combine $\frac{1}{3}$ and $x$ .

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

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Multiply $3$ by $- 1$ .

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

Let $u = \frac{x}{3}$ . Then $d u = \frac{1}{3} d x$ , so $3 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Let $u = \frac{x}{3}$ . Find $\frac{d u}{d x}$ .

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Differentiate $\frac{x}{3}$ .

$\frac{d}{d x} [\frac{x}{3}]$

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

$\frac{1}{3} \frac{d}{d x} [x]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{3} \cdot 1$

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3}$

Rewrite the problem using $u$ and $d u$ .

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

Simplify.

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Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

$3 x e^{\frac{x}{3}} - 3 \int e^{u} (1 \cdot 3) d u$

Multiply $3$ by $1$ .

$3 x e^{\frac{x}{3}} - 3 \int e^{u} \cdot 3 d u$

Move $3$ to the left of $e^{u}$ .

$3 x e^{\frac{x}{3}} - 3 \int 3 e^{u} d u$

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

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

Multiply $3$ by $- 3$ .

$3 x e^{\frac{x}{3}} - 9 \int e^{u} d u$

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$3 x e^{\frac{x}{3}} - 9 (e^{u} + C)$

Rewrite $3 x e^{\frac{x}{3}} - 9 (e^{u} + C)$ as $3 x e^{\frac{1}{3} x} - 9 e^{u} + C$ .

$3 x e^{\frac{1}{3} x} - 9 e^{u} + C$

Replace all occurrences of $u$ with $\frac{x}{3}$ .

$3 x e^{\frac{1}{3} x} - 9 e^{\frac{x}{3}} + C$

Reorder terms.

$3 x e^{\frac{1}{3} x} - 9 e^{\frac{1}{3} x} + C$

Evaluate integral of xe^(x/3) with respect to x

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