Calculus Examples

$\int e^{- x^{4}} (- 4 x^{3}) d x$

Step 1

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$- 4 \int e^{- x^{4}} (x^{3}) d x$

Step 2

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

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Step 2.1

Let $u_{1} = x^{2}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $x^{2}$ .

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

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x$

Step 2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$- 4 \int e^{- {\sqrt{u_{1}}}^{4}} u_{1} \frac{1}{2} d u_{1}$

Step 3

Simplify.

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Step 3.1

Rewrite ${\sqrt{u_{1}}}^{4}$ as ${u_{1}}^{2}$ .

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u_{1}}$ as ${u_{1}}^{\frac{1}{2}}$ .

$- 4 \int e^{- {({u_{1}}^{\frac{1}{2}})}^{4}} u_{1} \frac{1}{2} d u_{1}$

Step 3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 4 \int e^{- {u_{1}}^{\frac{1}{2} \cdot 4}} u_{1} \frac{1}{2} d u_{1}$

Step 3.1.3

Combine $\frac{1}{2}$ and $4$ .

$- 4 \int e^{- {u_{1}}^{\frac{4}{2}}} u_{1} \frac{1}{2} d u_{1}$

Step 3.1.4

Cancel the common factor of $4$ and $2$ .

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Step 3.1.4.1

Factor $2$ out of $4$ .

$- 4 \int e^{- {u_{1}}^{\frac{2 \cdot 2}{2}}} u_{1} \frac{1}{2} d u_{1}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $2$ .

$- 4 \int e^{- {u_{1}}^{\frac{2 \cdot 2}{2 (1)}}} u_{1} \frac{1}{2} d u_{1}$

Step 3.1.4.2.2

Cancel the common factor.

$- 4 \int e^{- {u_{1}}^{\frac{2 \cdot 2}{2 \cdot 1}}} u_{1} \frac{1}{2} d u_{1}$

Step 3.1.4.2.3

Rewrite the expression.

$- 4 \int e^{- {u_{1}}^{\frac{2}{1}}} u_{1} \frac{1}{2} d u_{1}$

Step 3.1.4.2.4

Divide $2$ by $1$ .

$- 4 \int e^{- {u_{1}}^{2}} u_{1} \frac{1}{2} d u_{1}$

Step 3.2

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

$- 4 \int \frac{e^{- {u_{1}}^{2}}}{2} u_{1} d u_{1}$

Step 3.3

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

$- 4 \int \frac{e^{- {u_{1}}^{2}} u_{1}}{2} d u_{1}$

Step 4

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

$- 4 (\frac{1}{2} \int e^{- {u_{1}}^{2}} u_{1} d u_{1})$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{2}$ and $- 4$ .

$\frac{- 4}{2} \int e^{- {u_{1}}^{2}} u_{1} d u_{1}$

Step 5.2

Cancel the common factor of $- 4$ and $2$ .

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Step 5.2.1

Factor $2$ out of $- 4$ .

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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

Step 5.2.2.4

Divide $- 2$ by $1$ .

$- 2 \int e^{- {u_{1}}^{2}} u_{1} d u_{1}$

Step 6

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

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Step 6.1

Let $u_{2} = - {u_{1}}^{2}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 6.1.1

Differentiate $- {u_{1}}^{2}$ .

$\frac{d}{d u_{1}} [- {u_{1}}^{2}]$

Step 6.1.2

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

$- \frac{d}{d u_{1}} [{u_{1}}^{2}]$

Step 6.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$- (2 u_{1})$

Step 6.1.4

Multiply $2$ by $- 1$ .

$- 2 u_{1}$

Step 6.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

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

Step 7

Simplify.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

$- 2 \int - \frac{e^{u_{2}}}{2} d u_{2}$

Step 8

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

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

Step 9

Multiply $- 1$ by $- 2$ .

$2 \int \frac{e^{u_{2}}}{2} d u_{2}$

Step 10

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

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

Step 11

Simplify.

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Step 11.1

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

$\frac{2}{2} \int e^{u_{2}} d u_{2}$

Step 11.2

Cancel the common factor of $2$ .

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Step 11.2.1

Cancel the common factor.

$\frac{2}{2} \int e^{u_{2}} d u_{2}$

Step 11.2.2

Rewrite the expression.

$1 \int e^{u_{2}} d u_{2}$

Step 11.3

Multiply $\int e^{u_{2}} d u_{2}$ by $1$ .

$\int e^{u_{2}} d u_{2}$

Step 12

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$e^{u_{2}} + C$

Step 13

Substitute back in for each integration substitution variable.

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Step 13.1

Replace all occurrences of $u_{2}$ with $- {u_{1}}^{2}$ .

$e^{- {u_{1}}^{2}} + C$

Step 13.2

Replace all occurrences of $u_{1}$ with $x^{2}$ .

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