Calculus Examples

$\frac{d y}{d x} + y = \sin (x)$

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 $e^{x}$ .

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

Step 2.2

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

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

Step 3

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

$\frac{d}{d x} [e^{x} y] = e^{x} \sin (x)$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{x} y] d x = \int e^{x} \sin (x) d x$

Step 5

Integrate the left side.

$e^{x} y = \int e^{x} \sin (x) d x$

Step 6

Integrate the right side.

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Step 6.1

Reorder $e^{x}$ and $\sin (x)$ .

$e^{x} y = \int \sin (x) e^{x} d x$

Step 6.2

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

$e^{x} y = \sin (x) e^{x} - \int e^{x} \cos (x) d x$

Step 6.3

Reorder $e^{x}$ and $\cos (x)$ .

$e^{x} y = \sin (x) e^{x} - \int \cos (x) e^{x} d x$

Step 6.4

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

$e^{x} y = \sin (x) e^{x} - (\cos (x) e^{x} - \int e^{x} (- \sin (x)) d x)$

Step 6.5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$e^{x} y = \sin (x) e^{x} - (\cos (x) e^{x} - - \int e^{x} (\sin (x)) d x)$

Step 6.6

Simplify by multiplying through.

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Step 6.6.1

Multiply $- 1$ by $- 1$ .

$e^{x} y = \sin (x) e^{x} - (\cos (x) e^{x} + 1 \int e^{x} (\sin (x)) d x)$

Step 6.6.2

Multiply $\int e^{x} (\sin (x)) d x$ by $1$ .

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

Step 6.6.3

Apply the distributive property.

$e^{x} y = \sin (x) e^{x} - (\cos (x) e^{x}) - \int e^{x} (\sin (x)) d x$

Step 6.7

Solving for $\int e^{x} \sin (x) d x$ , we find that $\int e^{x} \sin (x) d x$ = $\frac{\sin (x) e^{x} - (\cos (x) e^{x})}{2}$ .

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

Step 6.8

Rewrite $\frac{\sin (x) e^{x} - \cos (x) e^{x}}{2} + C$ as $\frac{1}{2} (\sin (x) e^{x} - \cos (x) e^{x}) + C$ .

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

Step 7

Solve for $y$ .

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Step 7.1

Simplify.

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Step 7.1.1

Apply the distributive property.

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

Step 7.1.2

Multiply $\frac{1}{2} (\sin (x) e^{x})$ .

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Step 7.1.2.1

Combine $\sin (x)$ and $\frac{1}{2}$ .

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

Step 7.1.2.2

Combine $\frac{\sin (x)}{2}$ and $e^{x}$ .

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

Step 7.1.3

Multiply $\frac{1}{2} (- \cos (x) e^{x})$ .

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Step 7.1.3.1

Combine $\frac{1}{2}$ and $\cos (x)$ .

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

Step 7.1.3.2

Combine $e^{x}$ and $\frac{\cos (x)}{2}$ .

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

Step 7.1.4

Reorder factors in $\frac{\sin (x) e^{x}}{2} - \frac{e^{x} \cos (x)}{2}$ .

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

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

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Step 7.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.1.2

Divide $y$ by $1$ .

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

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Simplify each term.

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Step 7.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2.3.1.2

Combine.

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

Step 7.2.3.1.3

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

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Step 7.2.3.1.3.1

Cancel the common factor.

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

Step 7.2.3.1.3.2

Rewrite the expression.

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

Step 7.2.3.1.4

Multiply $\sin (x)$ by $1$ .

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

Step 7.2.3.1.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2.3.1.6

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

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Step 7.2.3.1.6.1

Move the leading negative in $- \frac{e^{x} \cos (x)}{2}$ into the numerator.

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

Step 7.2.3.1.6.2

Factor $e^{x}$ out of $- e^{x} \cos (x)$ .

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

Step 7.2.3.1.6.3

Cancel the common factor.

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

Step 7.2.3.1.6.4

Rewrite the expression.

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

Step 7.2.3.1.7

Move the negative in front of the fraction.

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