Calculus Examples

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

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 1$ .

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Step 1.1

Set up the integration.

$e^{\int d x}$

Step 1.2

Apply the constant rule.

$e^{x + C}$

Step 1.3

Remove the constant of integration.

$e^{x}$

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

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Step 6.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{3}$ and $d v = e^{x}$ .

$e^{x} y = x^{3} e^{x} - \int e^{x} (3 x^{2}) d x$

Step 6.2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$e^{x} y = x^{3} e^{x} - (3 \int e^{x} (x^{2}) d x)$

Step 6.3

Multiply $3$ by $- 1$ .

$e^{x} y = x^{3} e^{x} - 3 \int e^{x} (x^{2}) d x$

Step 6.4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = e^{x}$ .

$e^{x} y = x^{3} e^{x} - 3 (x^{2} e^{x} - \int e^{x} (2 x) d x)$

Step 6.5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$e^{x} y = x^{3} e^{x} - 3 (x^{2} e^{x} - (2 \int e^{x} (x) d x))$

Step 6.6

Multiply $2$ by $- 1$ .

$e^{x} y = x^{3} e^{x} - 3 (x^{2} e^{x} - 2 \int e^{x} (x) d x)$

Step 6.7

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{x}$ .

$e^{x} y = x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (x e^{x} - \int e^{x} d x))$

Step 6.8

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

$e^{x} y = x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (x e^{x} - (e^{x} + C)))$

Step 6.9

Rewrite $x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (x e^{x} - (e^{x} + C)))$ as $x^{3} e^{x} - 3 (x^{2} e^{x} - 2 x e^{x} + 2 e^{x}) + C$ .

$e^{x} y = x^{3} e^{x} - 3 (x^{2} e^{x} - 2 x e^{x} + 2 e^{x}) + C$

Step 7

Divide each term in $e^{x} y = x^{3} e^{x} - 3 (x^{2} e^{x} - 2 x e^{x} + 2 e^{x}) + C$ by $e^{x}$ and simplify.

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Step 7.1

Divide each term in $e^{x} y = x^{3} e^{x} - 3 (x^{2} e^{x} - 2 x e^{x} + 2 e^{x}) + C$ by $e^{x}$ .

$\frac{e^{x} y}{e^{x}} = \frac{x^{3} e^{x}}{e^{x}} + \frac{- 3 (x^{2} e^{x} - 2 x e^{x} + 2 e^{x})}{e^{x}} + \frac{C}{e^{x}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{x} y}{e^{x}} = \frac{x^{3} e^{x}}{e^{x}} + \frac{- 3 (x^{2} e^{x} - 2 x e^{x} + 2 e^{x})}{e^{x}} + \frac{C}{e^{x}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{3} e^{x}}{e^{x}} + \frac{- 3 (x^{2} e^{x} - 2 x e^{x} + 2 e^{x})}{e^{x}} + \frac{C}{e^{x}}$

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 $e^{x}$ .

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Step 7.3.1.1.1

Cancel the common factor.

$y = \frac{x^{3} e^{x}}{e^{x}} + \frac{- 3 (x^{2} e^{x} - 2 x e^{x} + 2 e^{x})}{e^{x}} + \frac{C}{e^{x}}$

Step 7.3.1.1.2

Divide $x^{3}$ by $1$ .

$y = x^{3} + \frac{- 3 (x^{2} e^{x} - 2 x e^{x} + 2 e^{x})}{e^{x}} + \frac{C}{e^{x}}$

Step 7.3.1.2

Factor $e^{x}$ out of $x^{2} e^{x} - 2 x e^{x} + 2 e^{x}$ .

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Step 7.3.1.2.1

Factor $e^{x}$ out of $x^{2} e^{x}$ .

$y = x^{3} + \frac{- 3 (e^{x} x^{2} - 2 x e^{x} + 2 e^{x})}{e^{x}} + \frac{C}{e^{x}}$

Step 7.3.1.2.2

Factor $e^{x}$ out of $- 2 x e^{x}$ .

$y = x^{3} + \frac{- 3 (e^{x} x^{2} + e^{x} (- 2 x) + 2 e^{x})}{e^{x}} + \frac{C}{e^{x}}$

Step 7.3.1.2.3

Factor $e^{x}$ out of $2 e^{x}$ .

$y = x^{3} + \frac{- 3 (e^{x} x^{2} + e^{x} (- 2 x) + e^{x} \cdot 2)}{e^{x}} + \frac{C}{e^{x}}$

Step 7.3.1.2.4

Factor $e^{x}$ out of $e^{x} x^{2} + e^{x} (- 2 x)$ .

$y = x^{3} + \frac{- 3 (e^{x} (x^{2} - 2 x) + e^{x} \cdot 2)}{e^{x}} + \frac{C}{e^{x}}$

Step 7.3.1.2.5

Factor $e^{x}$ out of $e^{x} (x^{2} - 2 x) + e^{x} \cdot 2$ .

$y = x^{3} + \frac{- 3 (e^{x} (x^{2} - 2 x + 2))}{e^{x}} + \frac{C}{e^{x}}$

$y = x^{3} + \frac{- 3 e^{x} (x^{2} - 2 x + 2)}{e^{x}} + \frac{C}{e^{x}}$

Step 7.3.1.3

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

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Step 7.3.1.3.1

Cancel the common factor.

$y = x^{3} + \frac{- 3 e^{x} (x^{2} - 2 x + 2)}{e^{x}} + \frac{C}{e^{x}}$

Step 7.3.1.3.2

Divide $- 3 (x^{2} - 2 x + 2)$ by $1$ .

$y = x^{3} - 3 (x^{2} - 2 x + 2) + \frac{C}{e^{x}}$

Step 7.3.1.4

Apply the distributive property.

$y = x^{3} - 3 x^{2} - 3 (- 2 x) - 3 \cdot 2 + \frac{C}{e^{x}}$

Step 7.3.1.5

Simplify.

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Step 7.3.1.5.1

Multiply $- 2$ by $- 3$ .

$y = x^{3} - 3 x^{2} + 6 x - 3 \cdot 2 + \frac{C}{e^{x}}$

Step 7.3.1.5.2

Multiply $- 3$ by $2$ .

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