Calculus Examples

$\int 6 e^{- 7 x} d x$

Step 1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int e^{- 7 x} d x$

Step 2

Let $u = - 7 x$ . Then $d u = - 7 d x$ , so $- \frac{1}{7} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = - 7 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $- 7 x$ .

$\frac{d}{d x} [- 7 x]$

Step 2.1.2

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$- 7 \frac{d}{d x} [x]$

Step 2.1.3

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

$- 7 \cdot 1$

Step 2.1.4

Multiply $- 7$ by $1$ .

$- 7$

Step 2.2

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

$6 \int e^{u} \frac{1}{- 7} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$6 \int e^{u} (- \frac{1}{7}) d u$

Step 3.2

Combine $e^{u}$ and $\frac{1}{7}$ .

$6 \int - \frac{e^{u}}{7} d u$

Step 4

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

$6 (- \int \frac{e^{u}}{7} d u)$

Step 5

Multiply $- 1$ by $6$ .

$- 6 \int \frac{e^{u}}{7} d u$

Step 6

Since $\frac{1}{7}$ is constant with respect to $u$ , move $\frac{1}{7}$ out of the integral.

$- 6 (\frac{1}{7} \int e^{u} d u)$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{7}$ and $- 6$ .

$\frac{- 6}{7} \int e^{u} d u$

Step 7.2

Move the negative in front of the fraction.

$- \frac{6}{7} \int e^{u} d u$

Step 8

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

$- \frac{6}{7} (e^{u} + C)$

Step 9

Simplify.

$- \frac{6}{7} e^{u} + C$

Step 10

Replace all occurrences of $u$ with $- 7 x$ .

$- \frac{6}{7} e^{- 7 x} + C$