Calculus Examples

$\frac{d q}{d t} + q = 4 \cos (2 t)$ , $q (0) = 1$

Step 1

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

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Step 1.1

Set up the integration.

$e^{\int d t}$

Step 1.2

Apply the constant rule.

$e^{t + C}$

Step 1.3

Remove the constant of integration.

$e^{t}$

Step 2

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

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Step 2.1

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

$e^{t} \frac{d q}{d t} + e^{t} q = e^{t} (4 \cos (2 t))$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{t} \frac{d q}{d t} + e^{t} q = 4 e^{t} \cos (2 t)$

Step 2.3

Reorder factors in $e^{t} \frac{d q}{d t} + e^{t} q = 4 e^{t} \cos (2 t)$ .

$e^{t} \frac{d q}{d t} + q e^{t} = 4 e^{t} \cos (2 t)$

Step 3

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

$\frac{d}{d t} [e^{t} q] = 4 e^{t} \cos (2 t)$

Step 4

Set up an integral on each side.

$\int \frac{d}{d t} [e^{t} q] d t = \int 4 e^{t} \cos (2 t) d t$

Step 5

Integrate the left side.

$e^{t} q = \int 4 e^{t} \cos (2 t) d t$

Step 6

Integrate the right side.

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Step 6.1

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

$e^{t} q = 4 \int e^{t} \cos (2 t) d t$

Step 6.2

Reorder $e^{t}$ and $\cos (2 t)$ .

$e^{t} q = 4 \int \cos (2 t) e^{t} d t$

Step 6.3

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

$e^{t} q = 4 (\cos (2 t) e^{t} - \int e^{t} (- 2 \sin (2 t)) d t)$

Step 6.4

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

$e^{t} q = 4 (\cos (2 t) e^{t} - (- 2 \int e^{t} (\sin (2 t)) d t))$

Step 6.5

Simplify the expression.

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Step 6.5.1

Multiply $- 2$ by $- 1$ .

$e^{t} q = 4 (\cos (2 t) e^{t} + 2 \int e^{t} (\sin (2 t)) d t)$

Step 6.5.2

Reorder $e^{t}$ and $\sin (2 t)$ .

$e^{t} q = 4 (\cos (2 t) e^{t} + 2 \int \sin (2 t) e^{t} d t)$

Step 6.6

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

$e^{t} q = 4 (\cos (2 t) e^{t} + 2 (\sin (2 t) e^{t} - \int e^{t} (2 \cos (2 t)) d t))$

Step 6.7

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

$e^{t} q = 4 (\cos (2 t) e^{t} + 2 (\sin (2 t) e^{t} - (2 \int e^{t} (\cos (2 t)) d t)))$

Step 6.8

Simplify by multiplying through.

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Step 6.8.1

Multiply $2$ by $- 1$ .

$e^{t} q = 4 (\cos (2 t) e^{t} + 2 (\sin (2 t) e^{t} - 2 \int e^{t} (\cos (2 t)) d t))$

Step 6.8.2

Apply the distributive property.

$e^{t} q = 4 (\cos (2 t) e^{t} + 2 (\sin (2 t) e^{t}) + 2 (- 2 \int e^{t} (\cos (2 t)) d t))$

Step 6.8.3

Multiply $- 2$ by $2$ .

$e^{t} q = 4 (\cos (2 t) e^{t} + 2 (\sin (2 t) e^{t}) - 4 \int e^{t} (\cos (2 t)) d t)$

Step 6.9

Solving for $\int e^{t} \cos (2 t) d t$ , we find that $\int e^{t} \cos (2 t) d t$ = $\frac{\cos (2 t) e^{t} + 2 (\sin (2 t) e^{t})}{5}$ .

$e^{t} q = 4 (\frac{\cos (2 t) e^{t} + 2 (\sin (2 t) e^{t})}{5} + C)$

Step 6.10

Simplify the answer.

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Step 6.10.1

Rewrite $4 (\frac{\cos (2 t) e^{t} + 2 \sin (2 t) e^{t}}{5} + C)$ as $4 \frac{e^{t} (\cos (2 t) + 2 \sin (2 t))}{5} + C$ .

$e^{t} q = 4 \frac{e^{t} (\cos (2 t) + 2 \sin (2 t))}{5} + C$

Step 6.10.2

Simplify.

$e^{t} q = \frac{4}{5} e^{t} (\cos (2 t) + 2 \sin (2 t)) + C$

Step 7

Divide each term in $e^{t} q = \frac{4}{5} \cdot (e^{t} (\cos (2 t) + 2 \sin (2 t))) + C$ by $e^{t}$ and simplify.

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Step 7.1

Divide each term in $e^{t} q = \frac{4}{5} \cdot (e^{t} (\cos (2 t) + 2 \sin (2 t))) + C$ by $e^{t}$ .

$\frac{e^{t} q}{e^{t}} = \frac{\frac{4}{5} \cdot (e^{t} (\cos (2 t) + 2 \sin (2 t)))}{e^{t}} + \frac{C}{e^{t}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{t} q}{e^{t}} = \frac{\frac{4}{5} \cdot (e^{t} (\cos (2 t) + 2 \sin (2 t)))}{e^{t}} + \frac{C}{e^{t}}$

Step 7.2.1.2

Divide $q$ by $1$ .

$q = \frac{\frac{4}{5} \cdot (e^{t} (\cos (2 t) + 2 \sin (2 t)))}{e^{t}} + \frac{C}{e^{t}}$

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 $e^{t}$ .

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Step 7.3.1.1.1

Cancel the common factor.

$q = \frac{\frac{4}{5} \cdot (e^{t} (\cos (2 t) + 2 \sin (2 t)))}{e^{t}} + \frac{C}{e^{t}}$

Step 7.3.1.1.2

Divide $\frac{4}{5} \cdot (\cos (2 t) + 2 \sin (2 t))$ by $1$ .

$q = \frac{4}{5} \cdot (\cos (2 t) + 2 \sin (2 t)) + \frac{C}{e^{t}}$

Step 7.3.1.2

Apply the distributive property.

$q = \frac{4}{5} \cos (2 t) + \frac{4}{5} (2 \sin (2 t)) + \frac{C}{e^{t}}$

Step 7.3.1.3

Combine $\frac{4}{5}$ and $\cos (2 t)$ .

$q = \frac{4 \cos (2 t)}{5} + \frac{4}{5} (2 \sin (2 t)) + \frac{C}{e^{t}}$

Step 7.3.1.4

Multiply $\frac{4}{5} (2 \sin (2 t))$ .

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Step 7.3.1.4.1

Combine $2$ and $\frac{4}{5}$ .

$q = \frac{4 \cos (2 t)}{5} + \frac{2 \cdot 4}{5} \sin (2 t) + \frac{C}{e^{t}}$

Step 7.3.1.4.2

Multiply $2$ by $4$ .

$q = \frac{4 \cos (2 t)}{5} + \frac{8}{5} \sin (2 t) + \frac{C}{e^{t}}$

Step 7.3.1.4.3

Combine $\frac{8}{5}$ and $\sin (2 t)$ .

$q = \frac{4 \cos (2 t)}{5} + \frac{8 \sin (2 t)}{5} + \frac{C}{e^{t}}$

Step 8

Use the initial condition to find the value of $C$ by substituting $0$ for $t$ and $1$ for $q$ in $q = \frac{4 \cos (2 t)}{5} + \frac{8 \sin (2 t)}{5} + \frac{C}{e^{t}}$ .

$1 = \frac{4 \cos (2 \cdot 0)}{5} + \frac{8 \sin (2 \cdot 0)}{5} + \frac{C}{e^{0}}$

Step 9

Solve for $C$ .

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Step 9.1

Rewrite the equation as $\frac{4 \cos (2 \cdot 0)}{5} + \frac{8 \sin (2 \cdot 0)}{5} + \frac{C}{e^{0}} = 1$ .

$\frac{4 \cos (2 \cdot 0)}{5} + \frac{8 \sin (2 \cdot 0)}{5} + \frac{C}{e^{0}} = 1$

Step 9.2

Simplify the left side.

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Step 9.2.1

Simplify $\frac{4 \cos (2 \cdot 0)}{5} + \frac{8 \sin (2 \cdot 0)}{5} + \frac{C}{e^{0}}$ .

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Step 9.2.1.1

Combine the numerators over the common denominator.

$\frac{4 \cos (2 \cdot 0) + 8 \sin (2 \cdot 0)}{5} + \frac{C}{e^{0}} = 1$

Step 9.2.1.2

Simplify each term.

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Step 9.2.1.2.1

Multiply $2$ by $0$ .

$\frac{4 \cos (0) + 8 \sin (2 \cdot 0)}{5} + \frac{C}{e^{0}} = 1$

Step 9.2.1.2.2

The exact value of $\cos (0)$ is $1$ .

$\frac{4 \cdot 1 + 8 \sin (2 \cdot 0)}{5} + \frac{C}{e^{0}} = 1$

Step 9.2.1.2.3

Multiply $4$ by $1$ .

$\frac{4 + 8 \sin (2 \cdot 0)}{5} + \frac{C}{e^{0}} = 1$

Step 9.2.1.2.4

Multiply $2$ by $0$ .

$\frac{4 + 8 \sin (0)}{5} + \frac{C}{e^{0}} = 1$

Step 9.2.1.2.5

The exact value of $\sin (0)$ is $0$ .

$\frac{4 + 8 \cdot 0}{5} + \frac{C}{e^{0}} = 1$

Step 9.2.1.2.6

Multiply $8$ by $0$ .

$\frac{4 + 0}{5} + \frac{C}{e^{0}} = 1$

Step 9.2.1.3

Add $4$ and $0$ .

$\frac{4}{5} + \frac{C}{e^{0}} = 1$

Step 9.2.1.4

Simplify each term.

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Step 9.2.1.4.1

Anything raised to $0$ is $1$ .

$\frac{4}{5} + \frac{C}{1} = 1$

Step 9.2.1.4.2

Divide $C$ by $1$ .

$\frac{4}{5} + C = 1$

Step 9.3

Move all terms not containing $C$ to the right side of the equation.

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Step 9.3.1

Subtract $\frac{4}{5}$ from both sides of the equation.

$C = 1 - \frac{4}{5}$

Step 9.3.2

Write $1$ as a fraction with a common denominator.

$C = \frac{5}{5} - \frac{4}{5}$

Step 9.3.3

Combine the numerators over the common denominator.

$C = \frac{5 - 4}{5}$

Step 9.3.4

Subtract $4$ from $5$ .

$C = \frac{1}{5}$

Step 10

Substitute $\frac{1}{5}$ for $C$ in $q = \frac{4 \cos (2 t)}{5} + \frac{8 \sin (2 t)}{5} + \frac{C}{e^{t}}$ and simplify.

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Step 10.1

Substitute $\frac{1}{5}$ for $C$ .

$q = \frac{4 \cos (2 t)}{5} + \frac{8 \sin (2 t)}{5} + \frac{\frac{1}{5}}{e^{t}}$

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply the numerator by the reciprocal of the denominator.

$q = \frac{4 \cos (2 t)}{5} + \frac{8 \sin (2 t)}{5} + \frac{1}{5} \cdot \frac{1}{e^{t}}$

Step 10.2.2

Combine.

$q = \frac{4 \cos (2 t)}{5} + \frac{8 \sin (2 t)}{5} + \frac{1 \cdot 1}{5 e^{t}}$

Step 10.2.3

Multiply $1$ by $1$ .

$q = \frac{4 \cos (2 t)}{5} + \frac{8 \sin (2 t)}{5} + \frac{1}{5 e^{t}}$