Calculus Examples

$e^{- \frac{x}{2}}$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $- \frac{x}{2}$ .

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

Step 1.1.2

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{x}{2}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [x]$ .

$- \frac{1}{2} \frac{d}{d x} [x]$

Step 1.1.3

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

$- \frac{1}{2} \cdot 1$

Step 1.1.4

Multiply $- 1$ by $1$ .

$- \frac{1}{2}$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

Dividing two negative values results in a positive value.

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

Step 2.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

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

Step 2.3

Multiply $2$ by $1$ .

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

Step 2.4

Multiply $2$ by $- 1$ .

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

Step 3

Since $- 2$ is constant with respect to $u$ , move $- 2$ out of the integral.

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

Step 4

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

$- 2 (e^{u} + C)$

Step 5

Simplify.

$- 2 e^{u} + C$

Step 6

Replace all occurrences of $u$ with $- \frac{x}{2}$ .

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

Step 7

Reorder terms.

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