Calculus Examples

$\frac{d y}{d x} + y = x + 1$

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 $e^{x}$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

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Step 6.1

Split the single integral into multiple integrals.

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

Step 6.2

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

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

Step 6.3

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

$e^{x} y = x e^{x} - (e^{x} + C) + \int e^{x} d x$

Step 6.4

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

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

Step 6.5

Simplify.

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Step 6.5.1

Simplify.

$e^{x} y = x e^{x} - e^{x} + e^{x} + C$

Step 6.5.2

Simplify.

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Step 6.5.2.1

Add $- e^{x}$ and $e^{x}$ .

$e^{x} y = x e^{x} + 0 + C$

Step 6.5.2.2

Add $x e^{x}$ and $0$ .

$e^{x} y = x e^{x} + C$

Step 7

Divide each term in $e^{x} y = x e^{x} + C$ by $e^{x}$ and simplify.

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Step 7.1

Divide each term in $e^{x} y = x e^{x} + C$ by $e^{x}$ .

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

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.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

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

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Step 7.3.1.1

Cancel the common factor.

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

Step 7.3.1.2

Divide $x$ by $1$ .

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