Calculus Examples

$e^{2 x} d x$

Step 1

Write $e^{2 x} d x$ as a function.

$f (x) = e^{2 x} d x$

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.

$F (x) = \int e^{2 x} d x d x$

Step 4

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

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

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{2 x}$ .

$d (x (\frac{1}{2} e^{2 x}) - \int \frac{1}{2} e^{2 x} d x)$

Step 6

Simplify.

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Step 6.1

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

$d (x \frac{e^{2 x}}{2} - \int \frac{1}{2} e^{2 x} d x)$

Step 6.2

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

$d (\frac{x e^{2 x}}{2} - \int \frac{1}{2} e^{2 x} d x)$

Step 6.3

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

$d (\frac{x e^{2 x}}{2} - \int \frac{e^{2 x}}{2} d x)$

Step 7

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$d (\frac{x e^{2 x}}{2} - (\frac{1}{2} \int e^{2 x} d x))$

Step 8

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 8.1

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

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Step 8.1.1

Differentiate $2 x$ .

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

Step 8.1.2

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 8.1.3

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 8.1.4

Multiply $2$ by $1$ .

$2$

Step 8.2

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

$d (\frac{x e^{2 x}}{2} - \frac{1}{2} \int e^{u} \frac{1}{2} d u)$

Step 9

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

$d (\frac{x e^{2 x}}{2} - \frac{1}{2} \int \frac{e^{u}}{2} d u)$

Step 10

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$d (\frac{x e^{2 x}}{2} - \frac{1}{2} (\frac{1}{2} \int e^{u} d u))$

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$d (\frac{x e^{2 x}}{2} - \frac{1}{2 \cdot 2} \int e^{u} d u)$

Step 11.2

Multiply $2$ by $2$ .

$d (\frac{x e^{2 x}}{2} - \frac{1}{4} \int e^{u} d u)$

Step 12

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

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

Step 13

Rewrite $d (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u} + C))$ as $d (\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{u}) + C$ .

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

Step 14

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

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

Step 15

The answer is the antiderivative of the function $f (x) = e^{2 x} d x$ .

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