Calculus Examples

$\int (- x^{5} + 6 e^{- 2 x}) d x$

Step 1

Remove parentheses.

$\int - x^{5} + 6 e^{- 2 x} d x$

Step 2

Split the single integral into multiple integrals.

$\int - x^{5} d x + \int 6 e^{- 2 x} d x$

Step 3

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

$- \int x^{5} d x + \int 6 e^{- 2 x} d x$

Step 4

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$- (\frac{1}{6} x^{6} + C) + \int 6 e^{- 2 x} d x$

Step 5

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

$- (\frac{1}{6} x^{6} + C) + 6 \int e^{- 2 x} d x$

Step 6

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 6.1

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

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Step 6.1.1

Differentiate $- 2 x$ .

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

Step 6.1.2

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 6.1.3

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 6.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 6.2

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

$- (\frac{1}{6} x^{6} + C) + 6 \int e^{u} \frac{1}{- 2} d u$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{6}$ and $x^{6}$ .

$- (\frac{x^{6}}{6} + C) + 6 \int e^{u} \frac{1}{- 2} d u$

Step 7.2

Move the negative in front of the fraction.

$- (\frac{x^{6}}{6} + C) + 6 \int e^{u} (- \frac{1}{2}) d u$

Step 7.3

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

$- (\frac{x^{6}}{6} + C) + 6 \int - \frac{e^{u}}{2} d u$

Step 8

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

$- (\frac{x^{6}}{6} + C) + 6 (- \int \frac{e^{u}}{2} d u)$

Step 9

Multiply $- 1$ by $6$ .

$- (\frac{x^{6}}{6} + C) - 6 \int \frac{e^{u}}{2} d u$

Step 10

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

$- (\frac{x^{6}}{6} + C) - 6 (\frac{1}{2} \int e^{u} d u)$

Step 11

Simplify.

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Step 11.1

Combine $\frac{1}{2}$ and $- 6$ .

$- (\frac{x^{6}}{6} + C) + \frac{- 6}{2} \int e^{u} d u$

Step 11.2

Cancel the common factor of $- 6$ and $2$ .

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Step 11.2.1

Factor $2$ out of $- 6$ .

$- (\frac{x^{6}}{6} + C) + \frac{2 \cdot - 3}{2} \int e^{u} d u$

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $2$ out of $2$ .

$- (\frac{x^{6}}{6} + C) + \frac{2 \cdot - 3}{2 (1)} \int e^{u} d u$

Step 11.2.2.2

Cancel the common factor.

$- (\frac{x^{6}}{6} + C) + \frac{2 \cdot - 3}{2 \cdot 1} \int e^{u} d u$

Step 11.2.2.3

Rewrite the expression.

$- (\frac{x^{6}}{6} + C) + \frac{- 3}{1} \int e^{u} d u$

Step 11.2.2.4

Divide $- 3$ by $1$ .

$- (\frac{x^{6}}{6} + C) - 3 \int e^{u} d u$

Step 12

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

$- (\frac{x^{6}}{6} + C) - 3 (e^{u} + C)$

Step 13

Simplify.

$- \frac{1}{6} x^{6} - 3 e^{u} + C$

Step 14

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

$- \frac{1}{6} x^{6} - 3 e^{- 2 x} + C$