Calculus Examples

$\frac{d y}{d x} = - 4 x^{3} e^{- x^{4}}$

Step 1

Rewrite the equation.

$d y = - 4 x^{3} e^{- x^{4}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - 4 x^{3} e^{- x^{4}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - 4 x^{3} e^{- x^{4}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$y + C_{1} = - 4 \int x^{3} e^{- x^{4}} d x$

Step 2.3.2

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 2.3.2.1

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

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Step 2.3.2.1.1

Differentiate $x^{2}$ .

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

Step 2.3.2.1.2

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 2.3.2.2

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

$y + C_{1} = - 4 \int u_{1} e^{- {\sqrt{u_{1}}}^{4}} \frac{1}{2} d u_{1}$

Step 2.3.3

Simplify.

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Step 2.3.3.1

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

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Step 2.3.3.1.1

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

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

Step 2.3.3.1.2

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

$y + C_{1} = - 4 \int u_{1} e^{- {u_{1}}^{\frac{1}{2} \cdot 4}} \frac{1}{2} d u_{1}$

Step 2.3.3.1.3

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

$y + C_{1} = - 4 \int u_{1} e^{- {u_{1}}^{\frac{4}{2}}} \frac{1}{2} d u_{1}$

Step 2.3.3.1.4

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

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Step 2.3.3.1.4.1

Factor $2$ out of $4$ .

$y + C_{1} = - 4 \int u_{1} e^{- {u_{1}}^{\frac{2 \cdot 2}{2}}} \frac{1}{2} d u_{1}$

Step 2.3.3.1.4.2

Cancel the common factors.

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Step 2.3.3.1.4.2.1

Factor $2$ out of $2$ .

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

Step 2.3.3.1.4.2.2

Cancel the common factor.

$y + C_{1} = - 4 \int u_{1} e^{- {u_{1}}^{\frac{2 \cdot 2}{2 \cdot 1}}} \frac{1}{2} d u_{1}$

Step 2.3.3.1.4.2.3

Rewrite the expression.

$y + C_{1} = - 4 \int u_{1} e^{- {u_{1}}^{\frac{2}{1}}} \frac{1}{2} d u_{1}$

Step 2.3.3.1.4.2.4

Divide $2$ by $1$ .

$y + C_{1} = - 4 \int u_{1} e^{- {u_{1}}^{2}} \frac{1}{2} d u_{1}$

Step 2.3.3.2

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

$y + C_{1} = - 4 \int \frac{u_{1}}{2} e^{- {u_{1}}^{2}} d u_{1}$

Step 2.3.3.3

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

$y + C_{1} = - 4 \int \frac{u_{1} e^{- {u_{1}}^{2}}}{2} d u_{1}$

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

$y + C_{1} = \frac{- 4}{2} \int u_{1} e^{- {u_{1}}^{2}} d u_{1}$

Step 2.3.5.2

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

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Step 2.3.5.2.1

Factor $2$ out of $- 4$ .

$y + C_{1} = \frac{2 \cdot - 2}{2} \int u_{1} e^{- {u_{1}}^{2}} d u_{1}$

Step 2.3.5.2.2

Cancel the common factors.

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Step 2.3.5.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.5.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 \cdot - 2}{2 \cdot 1} \int u_{1} e^{- {u_{1}}^{2}} d u_{1}$

Step 2.3.5.2.2.3

Rewrite the expression.

$y + C_{1} = \frac{- 2}{1} \int u_{1} e^{- {u_{1}}^{2}} d u_{1}$

Step 2.3.5.2.2.4

Divide $- 2$ by $1$ .

$y + C_{1} = - 2 \int u_{1} e^{- {u_{1}}^{2}} d u_{1}$

Step 2.3.6

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 2.3.6.1

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

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Step 2.3.6.1.1

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

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

Step 2.3.6.1.2

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 2.3.6.1.3

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 2.3.6.1.4

Multiply $2$ by $- 1$ .

$- 2 u_{1}$

Step 2.3.6.2

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

$y + C_{1} = - 2 \int e^{u_{2}} \frac{1}{- 2} d u_{2}$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Move the negative in front of the fraction.

$y + C_{1} = - 2 \int e^{u_{2}} (- \frac{1}{2}) d u_{2}$

Step 2.3.7.2

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

$y + C_{1} = - 2 \int - \frac{e^{u_{2}}}{2} d u_{2}$

Step 2.3.8

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

$y + C_{1} = - 2 (- \int \frac{e^{u_{2}}}{2} d u_{2})$

Step 2.3.9

Multiply $- 1$ by $- 2$ .

$y + C_{1} = 2 \int \frac{e^{u_{2}}}{2} d u_{2}$

Step 2.3.10

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

$y + C_{1} = 2 (\frac{1}{2} \int e^{u_{2}} d u_{2})$

Step 2.3.11

Simplify.

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Step 2.3.11.1

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

$y + C_{1} = \frac{2}{2} \int e^{u_{2}} d u_{2}$

Step 2.3.11.2

Cancel the common factor of $2$ .

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Step 2.3.11.2.1

Cancel the common factor.

$y + C_{1} = \frac{2}{2} \int e^{u_{2}} d u_{2}$

Step 2.3.11.2.2

Rewrite the expression.

$y + C_{1} = 1 \int e^{u_{2}} d u_{2}$

Step 2.3.11.3

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

$y + C_{1} = \int e^{u_{2}} d u_{2}$

Step 2.3.12

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

$y + C_{1} = e^{u_{2}} + C_{2}$

Step 2.3.13

Substitute back in for each integration substitution variable.

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Step 2.3.13.1

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

$y + C_{1} = e^{- {u_{1}}^{2}} + C_{2}$

Step 2.3.13.2

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

$y + C_{1} = e^{- {(x^{2})}^{2}} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = e^{- {(x^{2})}^{2}} + K$