Calculus Examples

$\frac{d N}{d t} = 2 (1 - \cos (\frac{2 π t}{24})) N$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{N}$ .

$\frac{1}{N} \frac{d N}{d t} = \frac{1}{N} (2 (1 - \cos (\frac{2 π t}{24})) N)$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{N} \frac{d N}{d t} = 2 \frac{1}{N} ((1 - \cos (\frac{2 π t}{24})) N)$

Step 1.2.2

Combine $2$ and $\frac{1}{N}$ .

$\frac{1}{N} \frac{d N}{d t} = \frac{2}{N} ((1 - \cos (\frac{2 π t}{24})) N)$

Step 1.2.3

Cancel the common factor of $N$ .

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Step 1.2.3.1

Factor $N$ out of $(1 - \cos (\frac{2 π t}{24})) N$ .

$\frac{1}{N} \frac{d N}{d t} = \frac{2}{N} (N (1 - \cos (\frac{2 π t}{24})))$

Step 1.2.3.2

Cancel the common factor.

$\frac{1}{N} \frac{d N}{d t} = \frac{2}{N} (N (1 - \cos (\frac{2 π t}{24})))$

Step 1.2.3.3

Rewrite the expression.

$\frac{1}{N} \frac{d N}{d t} = 2 (1 - \cos (\frac{2 π t}{24}))$

Step 1.2.4

Cancel the common factor of $2$ and $24$ .

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Step 1.2.4.1

Factor $2$ out of $2 π t$ .

$\frac{1}{N} \frac{d N}{d t} = 2 (1 - \cos (\frac{2 (π t)}{24}))$

Step 1.2.4.2

Cancel the common factors.

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Step 1.2.4.2.1

Factor $2$ out of $24$ .

$\frac{1}{N} \frac{d N}{d t} = 2 (1 - \cos (\frac{2 (π t)}{2 \cdot 12}))$

Step 1.2.4.2.2

Cancel the common factor.

$\frac{1}{N} \frac{d N}{d t} = 2 (1 - \cos (\frac{2 (π t)}{2 \cdot 12}))$

Step 1.2.4.2.3

Rewrite the expression.

$\frac{1}{N} \frac{d N}{d t} = 2 (1 - \cos (\frac{π t}{12}))$

Step 1.2.5

Apply the distributive property.

$\frac{1}{N} \frac{d N}{d t} = 2 \cdot 1 + 2 (- \cos (\frac{π t}{12}))$

Step 1.2.6

Multiply $2$ by $1$ .

$\frac{1}{N} \frac{d N}{d t} = 2 + 2 (- \cos (\frac{π t}{12}))$

Step 1.2.7

Multiply $- 1$ by $2$ .

$\frac{1}{N} \frac{d N}{d t} = 2 - 2 \cos (\frac{π t}{12})$

Step 1.3

Rewrite the equation.

$\frac{1}{N} d N = (2 - 2 \cos (\frac{π t}{12})) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{N} d N = \int 2 - 2 \cos (\frac{π t}{12}) d t$

Step 2.2

The integral of $\frac{1}{N}$ with respect to $N$ is $\ln (| N |)$ .

$\ln (| N |) + C_{1} = \int 2 - 2 \cos (\frac{π t}{12}) d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$\ln (| N |) + C_{1} = \int 2 d t + \int - 2 \cos (\frac{π t}{12}) d t$

Step 2.3.2

Apply the constant rule.

$\ln (| N |) + C_{1} = 2 t + C_{2} + \int - 2 \cos (\frac{π t}{12}) d t$

Step 2.3.3

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

$\ln (| N |) + C_{1} = 2 t + C_{2} - 2 \int \cos (\frac{π t}{12}) d t$

Step 2.3.4

Let $u = \frac{π t}{12}$ . Then $d u = \frac{π}{12} d t$ , so $\frac{12}{π} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.4.1

Let $u = \frac{π t}{12}$ . Find $\frac{d u}{d t}$ .

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Step 2.3.4.1.1

Differentiate $\frac{π t}{12}$ .

$\frac{d}{d t} [\frac{π t}{12}]$

Step 2.3.4.1.2

Since $\frac{π}{12}$ is constant with respect to $t$ , the derivative of $\frac{π t}{12}$ with respect to $t$ is $\frac{π}{12} \frac{d}{d t} [t]$ .

$\frac{π}{12} \frac{d}{d t} [t]$

Step 2.3.4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$\frac{π}{12} \cdot 1$

Step 2.3.4.1.4

Multiply $\frac{π}{12}$ by $1$ .

$\frac{π}{12}$

Step 2.3.4.2

Rewrite the problem using $u$ and $d u$ .

$\ln (| N |) + C_{1} = 2 t + C_{2} - 2 \int \cos (u) \frac{1}{\frac{π}{12}} d u$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Multiply by the reciprocal of the fraction to divide by $\frac{π}{12}$ .

$\ln (| N |) + C_{1} = 2 t + C_{2} - 2 \int \cos (u) (1 \frac{12}{π}) d u$

Step 2.3.5.2

Multiply $\frac{12}{π}$ by $1$ .

$\ln (| N |) + C_{1} = 2 t + C_{2} - 2 \int \cos (u) \frac{12}{π} d u$

Step 2.3.5.3

Combine $\cos (u)$ and $\frac{12}{π}$ .

$\ln (| N |) + C_{1} = 2 t + C_{2} - 2 \int \frac{\cos (u) \cdot 12}{π} d u$

Step 2.3.5.4

Move $12$ to the left of $\cos (u)$ .

$\ln (| N |) + C_{1} = 2 t + C_{2} - 2 \int \frac{12 \cos (u)}{π} d u$

Step 2.3.6

Since $\frac{12}{π}$ is constant with respect to $u$ , move $\frac{12}{π}$ out of the integral.

$\ln (| N |) + C_{1} = 2 t + C_{2} - 2 (\frac{12}{π} \int \cos (u) d u)$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Combine $\frac{12}{π}$ and $- 2$ .

$\ln (| N |) + C_{1} = 2 t + C_{2} + \frac{12 \cdot - 2}{π} \int \cos (u) d u$

Step 2.3.7.2

Multiply $12$ by $- 2$ .

$\ln (| N |) + C_{1} = 2 t + C_{2} + \frac{- 24}{π} \int \cos (u) d u$

Step 2.3.7.3

Move the negative in front of the fraction.

$\ln (| N |) + C_{1} = 2 t + C_{2} - \frac{24}{π} \int \cos (u) d u$

Step 2.3.8

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\ln (| N |) + C_{1} = 2 t + C_{2} - \frac{24}{π} (\sin (u) + C_{3})$

Step 2.3.9

Simplify.

$\ln (| N |) + C_{1} = 2 t - \frac{24}{π} \sin (u) + C_{4}$

Step 2.3.10

Replace all occurrences of $u$ with $\frac{π t}{12}$ .

$\ln (| N |) + C_{1} = 2 t - \frac{24}{π} \sin (\frac{π t}{12}) + C_{4}$

Step 2.3.11

Reorder terms.

$\ln (| N |) + C_{1} = 2 t - \frac{24}{π} \sin (\frac{π}{12} t) + C_{4}$

Step 2.3.12

Reorder terms.

$\ln (| N |) + C_{1} = - \frac{24}{π} \sin (\frac{π}{12} t) + 2 t + C_{4}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$\ln (| N |) = - \frac{24}{π} \sin (\frac{π}{12} t) + 2 t + C$

Step 3

Solve for $N$ .

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Step 3.1

To solve for $N$ , rewrite the equation using properties of logarithms.

$e^{\ln (| N |)} = e^{- \frac{24}{π} \sin (\frac{π}{12} t) + 2 t + C}$

Step 3.2

Rewrite $\ln (| N |) = - \frac{24}{π} \sin (\frac{π}{12} t) + 2 t + C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- \frac{24}{π} \sin (\frac{π}{12} t) + 2 t + C} = | N |$

Step 3.3

Solve for $N$ .

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Step 3.3.1

Rewrite the equation as $| N | = e^{- \frac{24}{π} \cdot \sin (\frac{π}{12} t) + 2 t + C}$ .

$| N | = e^{- \frac{24}{π} \cdot \sin (\frac{π}{12} t) + 2 t + C}$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Combine $\sin (\frac{π}{12} t)$ and $\frac{24}{π}$ .

$| N | = e^{- \frac{\sin (\frac{π}{12} t) \cdot 24}{π} + 2 t + C}$

Step 3.3.2.2

Move $24$ to the left of $\sin (\frac{π}{12} t)$ .

$| N | = e^{- \frac{24 \sin (\frac{π}{12} t)}{π} + 2 t + C}$

Step 3.3.3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$N = \pm e^{- \frac{24 \sin (\frac{π}{12} t)}{π} + 2 t + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{- \frac{24 \sin (\frac{π}{12} t)}{π} + 2 t + C}$ as $e^{- \frac{24 \sin (\frac{π}{12} t)}{π} + 2 t} e^{C}$ .

$N = \pm e^{- \frac{24 \sin (\frac{π}{12} t)}{π} + 2 t} e^{C}$

Step 4.2

Reorder $e^{- \frac{24 \sin (\frac{π}{12} t)}{π} + 2 t}$ and $e^{C}$ .

$N = \pm e^{C} e^{- \frac{24 \sin (\frac{π}{12} t)}{π} + 2 t}$

Step 4.3

Combine constants with the plus or minus.

$N = C e^{- \frac{24 \sin (\frac{π}{12} t)}{π} + 2 t}$