Calculus Examples

$\int x e^{2 x} d x$

Step 1

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 1.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 1.1.3

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

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

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

$\int \frac{u}{2} e^{u} \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

Combine $\frac{u}{2}$ and $e^{u}$ .

$\int \frac{u e^{u}}{2} \cdot \frac{1}{2} d u$

Step 2.2

Multiply $\frac{u e^{u}}{2}$ by $\frac{1}{2}$ .

$\int \frac{u e^{u}}{2 \cdot 2} d u$

Step 2.3

Multiply $2$ by $2$ .

$\int \frac{u e^{u}}{4} d u$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int u e^{u} d u$

Step 4

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

$\frac{1}{4} (u e^{u} - \int e^{u} d u)$

Step 5

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

$\frac{1}{4} (u e^{u} - (e^{u} + C))$

Step 6

Simplify.

$\frac{1}{4} (u e^{u} - e^{u}) + C$

Step 7

Replace all occurrences of $u$ with $2 x$ .

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

Factor $2$ out of $4$ .

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

Step 8.2.2

Factor $2$ out of $2 x e^{2 x}$ .

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

Step 8.2.3

Cancel the common factor.

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

Step 8.2.4

Rewrite the expression.

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

Step 8.3

Combine $x$ and $\frac{1}{2}$ .

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

Step 8.4

Combine $\frac{x}{2}$ and $e^{2 x}$ .

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

Step 8.5

Combine $\frac{1}{4}$ and $e^{2 x}$ .

$\frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4} + C$

Step 9

Reorder terms.

$\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{2 x} + C$