Calculus Examples

$x \cdot e^{\frac{x}{3}}$

Step 1

Write $x \cdot e^{\frac{x}{3}}$ as a function.

$f (x) = x \cdot e^{\frac{x}{3}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x \cdot e^{\frac{x}{3}} d x$

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Multiply $3$ by $- 1$ .

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

Step 7

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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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.1.3

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

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

Step 7.1.4

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

$\frac{1}{3}$

Step 7.2

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

Multiply $3$ by $1$ .

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

Step 8.3

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

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

Step 9

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

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

Step 10

Multiply $3$ by $- 3$ .

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Reorder terms.

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

Step 15

The answer is the antiderivative of the function $f (x) = x \cdot e^{\frac{x}{3}}$ .

$F (x) =$ $3 x e^{\frac{1}{3} x} - 9 e^{\frac{1}{3} x} + C$