Calculus Examples

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

Step 1

Subtract $y$ from both sides of the equation.

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

Step 2

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

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Step 2.1

Set up the integration.

$e^{\int - 1 d x}$

Step 2.2

Apply the constant rule.

$e^{- x + C}$

Step 2.3

Remove the constant of integration.

$e^{- x}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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 4

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

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

Step 5

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 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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

Step 7.2

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 7.3

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

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

Step 7.4

Simplify the expression.

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Step 7.4.1

Multiply $- 1$ by $- 1$ .

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

Step 7.4.2

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

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

Step 7.4.3

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

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

Step 7.5

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 7.6

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

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

Step 7.7

Multiply.

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Step 7.7.1

Multiply $- 1$ by $- 1$ .

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

Step 7.7.2

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 7.8

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 7.9

Simplify the answer.

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Step 7.9.1

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

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

Step 7.9.2

Simplify.

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Step 7.9.2.1

Simplify the numerator.

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Step 7.9.2.1.1

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

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Step 7.9.2.1.1.1

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

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

Step 7.9.2.1.1.2

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

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

Step 7.9.2.1.1.3

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

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

Step 7.9.2.1.2

Factor $- 1$ out of $- \sin (x) - \cos (x)$ .

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Step 7.9.2.1.2.1

Factor $- 1$ out of $- \sin (x)$ .

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

Step 7.9.2.1.2.2

Factor $- 1$ out of $- \cos (x)$ .

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

Step 7.9.2.1.2.3

Factor $- 1$ out of $- (\sin (x)) - (\cos (x))$ .

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

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

Step 7.9.2.1.3

Factor out negative.

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

Step 7.9.2.2

Move the negative in front of the fraction.

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

Step 8

Solve for $y$ .

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Step 8.1

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

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

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

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Step 8.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.1.2

Divide $y$ by $1$ .

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

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Simplify each term.

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Step 8.2.3.1.1

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

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

Step 8.2.3.1.2

Combine.

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

Step 8.2.3.1.3

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

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Step 8.2.3.1.3.1

Cancel the common factor.

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

Step 8.2.3.1.3.2

Rewrite the expression.

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

Step 8.2.3.1.4

Move the negative in front of the fraction.

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