Calculus Examples

$e^{- 5 x}$

Let $u = - 5 x$ . Then $d u = - 5 d x$ , so $- \frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Let $u = - 5 x$ . Find $\frac{d u}{d x}$ .

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Differentiate $- 5 x$ .

$\frac{d}{d x} [- 5 x]$

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$- 5 \frac{d}{d x} [x]$

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

$- 5 \cdot 1$

Multiply $- 5$ by $1$ .

$- 5$

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

$\int e^{u} \frac{1}{- 5} d u$

Simplify.

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Move the negative in front of the fraction.

$\int e^{u} (- \frac{1}{5}) d u$

Combine $e^{u}$ and $\frac{1}{5}$ .

$\int - \frac{e^{u}}{5} d u$

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

$- \int \frac{e^{u}}{5} d u$

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$- (\frac{1}{5} \int e^{u} d u)$

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

$- \frac{1}{5} (e^{u} + C)$

Simplify.

$- \frac{1}{5} e^{u} + C$

Replace all occurrences of $u$ with $- 5 x$ .

$- \frac{1}{5} e^{- 5 x} + C$