Calculus Examples

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

Step 1

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

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

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $- 2 x$ .

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

Step 2.1.2

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 2.1.3

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 2.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

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

Step 3.2

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

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

Step 4

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Multiply $2$ by $2$ .

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

Step 8

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

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

Step 9

Simplify.

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

Step 10

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

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