Calculus Examples

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

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 4 x^{3}$ .

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Step 1.1

Set up the integration.

$e^{\int 4 x^{3} d x}$

Step 1.2

Integrate $4 x^{3}$ .

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Step 1.2.1

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

$e^{4 \int x^{3} d x}$

Step 1.2.2

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$e^{4 (\frac{1}{4} x^{4} + C)}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Rewrite $4 (\frac{1}{4} x^{4} + C)$ as $4 (\frac{1}{4}) x^{4} + C$ .

$e^{4 (\frac{1}{4}) x^{4} + C}$

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

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

$e^{\frac{4}{4} x^{4} + C}$

Step 1.2.3.2.2

Cancel the common factor of $4$ .

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Step 1.2.3.2.2.1

Cancel the common factor.

$e^{\frac{4}{4} x^{4} + C}$

Step 1.2.3.2.2.2

Rewrite the expression.

$e^{1 x^{4} + C}$

Step 1.2.3.2.3

Multiply $x^{4}$ by $1$ .

$e^{x^{4} + C}$

Step 1.3

Remove the constant of integration.

$e^{x^{4}}$

Step 2

Multiply each term by the integrating factor $e^{x^{4}}$ .

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Step 2.1

Multiply each term by $e^{x^{4}}$ .

$e^{x^{4}} \frac{d y}{d x} + e^{x^{4}} (4 x^{3} y) = e^{x^{4}} x^{3}$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{x^{4}} \frac{d y}{d x} + 4 e^{x^{4}} (x^{3} y) = e^{x^{4}} x^{3}$

Step 2.3

Reorder factors in $e^{x^{4}} \frac{d y}{d x} + 4 e^{x^{4}} (x^{3} y) = e^{x^{4}} x^{3}$ .

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

Step 3

Rewrite the left side as a result of differentiating a product.

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

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{x^{4}} y] d x = \int x^{3} e^{x^{4}} d x$

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

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Step 6.1

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 6.1.1

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

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Step 6.1.1.1

Differentiate $x^{2}$ .

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

Step 6.1.1.2

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 6.1.2

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

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

Step 6.2

Simplify.

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Step 6.2.1

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

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Step 6.2.1.1

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

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

Step 6.2.1.2

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

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

Step 6.2.1.3

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

$e^{x^{4}} y = \int u_{1} e^{{u_{1}}^{\frac{4}{2}}} \frac{1}{2} d u_{1}$

Step 6.2.1.4

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

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Step 6.2.1.4.1

Factor $2$ out of $4$ .

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

Step 6.2.1.4.2

Cancel the common factors.

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Step 6.2.1.4.2.1

Factor $2$ out of $2$ .

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

Step 6.2.1.4.2.2

Cancel the common factor.

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

Step 6.2.1.4.2.3

Rewrite the expression.

$e^{x^{4}} y = \int u_{1} e^{{u_{1}}^{\frac{2}{1}}} \frac{1}{2} d u_{1}$

Step 6.2.1.4.2.4

Divide $2$ by $1$ .

$e^{x^{4}} y = \int u_{1} e^{{u_{1}}^{2}} \frac{1}{2} d u_{1}$

Step 6.2.2

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

$e^{x^{4}} y = \int \frac{u_{1}}{2} e^{{u_{1}}^{2}} d u_{1}$

Step 6.2.3

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

$e^{x^{4}} y = \int \frac{u_{1} e^{{u_{1}}^{2}}}{2} d u_{1}$

Step 6.3

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

$e^{x^{4}} y = \frac{1}{2} \int u_{1} e^{{u_{1}}^{2}} d u_{1}$

Step 6.4

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 6.4.1

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

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Step 6.4.1.1

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

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

Step 6.4.1.2

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 6.4.2

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

$e^{x^{4}} y = \frac{1}{2} \int e^{u_{2}} \frac{1}{2} d u_{2}$

Step 6.5

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

$e^{x^{4}} y = \frac{1}{2} \int \frac{e^{u_{2}}}{2} d u_{2}$

Step 6.6

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

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

Step 6.7

Simplify.

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Step 6.7.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 6.7.2

Multiply $2$ by $2$ .

$e^{x^{4}} y = \frac{1}{4} \int e^{u_{2}} d u_{2}$

Step 6.8

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

$e^{x^{4}} y = \frac{1}{4} (e^{u_{2}} + C)$

Step 6.9

Simplify.

$e^{x^{4}} y = \frac{1}{4} e^{u_{2}} + C$

Step 6.10

Substitute back in for each integration substitution variable.

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Step 6.10.1

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

$e^{x^{4}} y = \frac{1}{4} e^{{u_{1}}^{2}} + C$

Step 6.10.2

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

$e^{x^{4}} y = \frac{1}{4} e^{{(x^{2})}^{2}} + C$

Step 7

Divide each term in $e^{x^{4}} y = \frac{1}{4} e^{{(x^{2})}^{2}} + C$ by $e^{x^{4}}$ and simplify.

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Step 7.1

Divide each term in $e^{x^{4}} y = \frac{1}{4} e^{{(x^{2})}^{2}} + C$ by $e^{x^{4}}$ .

$\frac{e^{x^{4}} y}{e^{x^{4}}} = \frac{\frac{1}{4} e^{{(x^{2})}^{2}}}{e^{x^{4}}} + \frac{C}{e^{x^{4}}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $e^{x^{4}}$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{x^{4}} y}{e^{x^{4}}} = \frac{\frac{1}{4} e^{{(x^{2})}^{2}}}{e^{x^{4}}} + \frac{C}{e^{x^{4}}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{4} e^{{(x^{2})}^{2}}}{e^{x^{4}}} + \frac{C}{e^{x^{4}}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Cancel the common factor of $e^{{(x^{2})}^{2}}$ and $e^{x^{4}}$ .

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Step 7.3.1.1.1

Factor $e^{x^{4}}$ out of $\frac{1}{4} e^{{(x^{2})}^{2}}$ .

$y = \frac{e^{x^{4}} (\frac{1}{4} e^{{(x^{2})}^{2} - x^{4}})}{e^{x^{4}}} + \frac{C}{e^{x^{4}}}$

Step 7.3.1.1.2

Cancel the common factors.

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Step 7.3.1.1.2.1

Multiply by $1$ .

$y = \frac{e^{x^{4}} (\frac{1}{4} e^{{(x^{2})}^{2} - x^{4}})}{e^{x^{4}} \cdot 1} + \frac{C}{e^{x^{4}}}$

Step 7.3.1.1.2.2

Cancel the common factor.

$y = \frac{e^{x^{4}} (\frac{1}{4} e^{{(x^{2})}^{2} - x^{4}})}{e^{x^{4}} \cdot 1} + \frac{C}{e^{x^{4}}}$

Step 7.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{4} e^{{(x^{2})}^{2} - x^{4}}}{1} + \frac{C}{e^{x^{4}}}$

Step 7.3.1.1.2.4

Divide $\frac{1}{4} e^{{(x^{2})}^{2} - x^{4}}$ by $1$ .

$y = \frac{1}{4} e^{{(x^{2})}^{2} - x^{4}} + \frac{C}{e^{x^{4}}}$

Step 7.3.1.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 7.3.1.2.1

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

$y = \frac{1}{4} e^{x^{2 \cdot 2} - x^{4}} + \frac{C}{e^{x^{4}}}$

Step 7.3.1.2.2

Multiply $2$ by $2$ .

$y = \frac{1}{4} e^{x^{4} - x^{4}} + \frac{C}{e^{x^{4}}}$

Step 7.3.1.3

Subtract $x^{4}$ from $x^{4}$ .

$y = \frac{1}{4} e^{0} + \frac{C}{e^{x^{4}}}$

Step 7.3.1.4

Anything raised to $0$ is $1$ .

$y = \frac{1}{4} \cdot 1 + \frac{C}{e^{x^{4}}}$

Step 7.3.1.5

Multiply $\frac{1}{4}$ by $1$ .

$y = \frac{1}{4} + \frac{C}{e^{x^{4}}}$