Calculus Examples

$\frac{d u}{d t} = 2.4 - \frac{2 u}{25}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{2.4 - \frac{2 u}{25}}$ .

$\frac{1}{2.4 - \frac{2 u}{25}} \frac{d u}{d t} = \frac{1}{2.4 - \frac{2 u}{25}} (2.4 - \frac{2 u}{25})$

Step 1.2

Cancel the common factor of $2.4 - \frac{2 u}{25}$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{2.4 - \frac{2 u}{25}} \frac{d u}{d t} = \frac{1}{2.4 - \frac{2 u}{25}} (2.4 - \frac{2 u}{25})$

Step 1.2.2

Rewrite the expression.

$\frac{1}{2.4 - \frac{2 u}{25}} \frac{d u}{d t} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{2.4 - \frac{2 u}{25}} d u = 1 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{2.4 - \frac{2 u}{25}} d u = \int d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u_{1} = 2.4 - \frac{2 u}{25}$ . Then $d u_{1} = - \frac{2}{25} d u$ , so $- \frac{25}{2} d u_{1} = d u$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.2.1.1

Let $u_{1} = 2.4 - \frac{2 u}{25}$ . Find $\frac{d u_{1}}{d u}$ .

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Step 2.2.1.1.1

Differentiate $2.4 - \frac{2 u}{25}$ .

$\frac{d}{d u} [2.4 - \frac{2 u}{25}]$

Step 2.2.1.1.2

Differentiate.

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Step 2.2.1.1.2.1

By the Sum Rule, the derivative of $2.4 - \frac{2 u}{25}$ with respect to $u$ is $\frac{d}{d u} [2.4] + \frac{d}{d u} [- \frac{2 u}{25}]$ .

$\frac{d}{d u} [2.4] + \frac{d}{d u} [- \frac{2 u}{25}]$

Step 2.2.1.1.2.2

Since $2.4$ is constant with respect to $u$ , the derivative of $2.4$ with respect to $u$ is $0$ .

$0 + \frac{d}{d u} [- \frac{2 u}{25}]$

Step 2.2.1.1.3

Evaluate $\frac{d}{d u} [- \frac{2 u}{25}]$ .

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Step 2.2.1.1.3.1

Since $- \frac{2}{25}$ is constant with respect to $u$ , the derivative of $- \frac{2 u}{25}$ with respect to $u$ is $- \frac{2}{25} \frac{d}{d u} [u]$ .

$0 - \frac{2}{25} \frac{d}{d u} [u]$

Step 2.2.1.1.3.2

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

$0 - \frac{2}{25} \cdot 1$

Step 2.2.1.1.3.3

Multiply $- 1$ by $1$ .

$0 - \frac{2}{25}$

Step 2.2.1.1.4

Subtract $\frac{2}{25}$ from $0$ .

$- \frac{2}{25}$

Step 2.2.1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{- 1}{u_{1}} \cdot \frac{- 1}{- \frac{2}{25}} d u_{1} = \int d t$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Move the negative in front of the fraction.

$\int - \frac{1}{u_{1}} \cdot \frac{- 1}{- \frac{2}{25}} d u_{1} = \int d t$

Step 2.2.2.2

Dividing two negative values results in a positive value.

$\int - \frac{1}{u_{1}} \cdot \frac{1}{\frac{2}{25}} d u_{1} = \int d t$

Step 2.2.2.3

Multiply by the reciprocal of the fraction to divide by $\frac{2}{25}$ .

$\int - \frac{1}{u_{1}} (1 (\frac{25}{2})) d u_{1} = \int d t$

Step 2.2.2.4

Multiply $\frac{25}{2}$ by $1$ .

$\int - \frac{1}{u_{1}} \cdot \frac{25}{2} d u_{1} = \int d t$

Step 2.2.2.5

Multiply $\frac{25}{2}$ by $\frac{1}{u_{1}}$ .

$\int - \frac{25}{2 u_{1}} d u_{1} = \int d t$

Step 2.2.3

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

$- \int \frac{25}{2 u_{1}} d u_{1} = \int d t$

Step 2.2.4

Since $\frac{25}{2}$ is constant with respect to $u_{1}$ , move $\frac{25}{2}$ out of the integral.

$- (\frac{25}{2} \int \frac{1}{u_{1}} d u_{1}) = \int d t$

Step 2.2.5

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

$- \frac{25}{2} (\ln (| u_{1} |) + C_{1}) = \int d t$

Step 2.2.6

Simplify.

$- \frac{25}{2} \ln (| u_{1} |) + C_{1} = \int d t$

Step 2.2.7

Replace all occurrences of $u_{1}$ with $2.4 - \frac{2 u}{25}$ .

$- \frac{25}{2} \ln (| 2.4 - \frac{2 u}{25} |) + C_{1} = \int d t$

Step 2.2.8

Reorder terms.

$- \frac{25}{2} \ln (| 2.4 - \frac{2}{25} u |) + C_{1} = \int d t$

Step 2.3

Apply the constant rule.

$- \frac{25}{2} \ln (| 2.4 - \frac{2}{25} u |) + C_{1} = t + C_{2}$

Step 2.4

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

$- \frac{25}{2} \ln (| 2.4 - \frac{2}{25} u |) = t + C$

Step 3

Solve for $u$ .

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Step 3.1

Multiply both sides of the equation by $- \frac{2}{25}$ .

$- \frac{2}{25} (- \frac{25}{2} \ln (| 2.4 - \frac{2}{25} u |)) = - \frac{2}{25} (t + C)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- \frac{2}{25} (- \frac{25}{2} \ln (| 2.4 - \frac{2}{25} u |))$ .

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Step 3.2.1.1.1

Simplify each term.

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Step 3.2.1.1.1.1

Combine $u$ and $\frac{2}{25}$ .

$- \frac{2}{25} (- \frac{25}{2} \ln (| 2.4 - \frac{u \cdot 2}{25} |)) = - \frac{2}{25} (t + C)$

Step 3.2.1.1.1.2

Move $2$ to the left of $u$ .

$- \frac{2}{25} (- \frac{25}{2} \ln (| 2.4 - \frac{2 u}{25} |)) = - \frac{2}{25} (t + C)$

Step 3.2.1.1.2

Simplify terms.

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Step 3.2.1.1.2.1

Combine $\ln (| 2.4 - \frac{2 u}{25} |)$ and $\frac{25}{2}$ .

$- \frac{2}{25} (- \frac{\ln (| 2.4 - \frac{2 u}{25} |) \cdot 25}{2}) = - \frac{2}{25} (t + C)$

Step 3.2.1.1.2.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.2.1

Move the leading negative in $- \frac{2}{25}$ into the numerator.

$\frac{- 2}{25} (- \frac{\ln (| 2.4 - \frac{2 u}{25} |) \cdot 25}{2}) = - \frac{2}{25} (t + C)$

Step 3.2.1.1.2.2.2

Move the leading negative in $- \frac{\ln (| 2.4 - \frac{2 u}{25} |) \cdot 25}{2}$ into the numerator.

$\frac{- 2}{25} \cdot \frac{- \ln (| 2.4 - \frac{2 u}{25} |) \cdot 25}{2} = - \frac{2}{25} (t + C)$

Step 3.2.1.1.2.2.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{25} \cdot \frac{- \ln (| 2.4 - \frac{2 u}{25} |) \cdot 25}{2} = - \frac{2}{25} (t + C)$

Step 3.2.1.1.2.2.4

Cancel the common factor.

$\frac{2 \cdot - 1}{25} \cdot \frac{- \ln (| 2.4 - \frac{2 u}{25} |) \cdot 25}{2} = - \frac{2}{25} (t + C)$

Step 3.2.1.1.2.2.5

Rewrite the expression.

$\frac{- 1}{25} (- \ln (| 2.4 - \frac{2 u}{25} |) \cdot 25) = - \frac{2}{25} (t + C)$

Step 3.2.1.1.2.3

Cancel the common factor of $25$ .

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Step 3.2.1.1.2.3.1

Factor $25$ out of $- \ln (| 2.4 - \frac{2 u}{25} |) \cdot 25$ .

$\frac{- 1}{25} (25 \cdot (- \ln (| 2.4 - \frac{2 u}{25} |))) = - \frac{2}{25} (t + C)$

Step 3.2.1.1.2.3.2

Cancel the common factor.

$\frac{- 1}{25} (25 \cdot (- \ln (| 2.4 - \frac{2 u}{25} |))) = - \frac{2}{25} (t + C)$

Step 3.2.1.1.2.3.3

Rewrite the expression.

$- - \ln (| 2.4 - \frac{2 u}{25} |) = - \frac{2}{25} (t + C)$

Step 3.2.1.1.2.4

Multiply.

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Step 3.2.1.1.2.4.1

Multiply $- 1$ by $- 1$ .

$1 \ln (| 2.4 - \frac{2 u}{25} |) = - \frac{2}{25} (t + C)$

Step 3.2.1.1.2.4.2

Multiply $\ln (| 2.4 - \frac{2 u}{25} |)$ by $1$ .

$\ln (| 2.4 - \frac{2 u}{25} |) = - \frac{2}{25} (t + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- \frac{2}{25} (t + C)$ .

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Step 3.2.2.1.1

Simplify terms.

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Step 3.2.2.1.1.1

Apply the distributive property.

$\ln (| 2.4 - \frac{2 u}{25} |) = - \frac{2}{25} t - \frac{2}{25} C$

Step 3.2.2.1.1.2

Combine $t$ and $\frac{2}{25}$ .

$\ln (| 2.4 - \frac{2 u}{25} |) = - \frac{t \cdot 2}{25} - \frac{2}{25} C$

Step 3.2.2.1.1.3

Combine $C$ and $\frac{2}{25}$ .

$\ln (| 2.4 - \frac{2 u}{25} |) = - \frac{t \cdot 2}{25} - \frac{C \cdot 2}{25}$

Step 3.2.2.1.2

Simplify each term.

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Step 3.2.2.1.2.1

Move $2$ to the left of $t$ .

$\ln (| 2.4 - \frac{2 u}{25} |) = - \frac{2 \cdot t}{25} - \frac{C \cdot 2}{25}$

Step 3.2.2.1.2.2

Move $2$ to the left of $C$ .

$\ln (| 2.4 - \frac{2 u}{25} |) = - \frac{2 t}{25} - \frac{2 C}{25}$

Step 3.3

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

$e^{\ln (| 2.4 - \frac{2 u}{25} |)} = e^{- \frac{2 t}{25} - \frac{2 C}{25}}$

Step 3.4

Rewrite $\ln (| 2.4 - \frac{2 u}{25} |) = - \frac{2 t}{25} - \frac{2 C}{25}$ 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 $b^{y} = x$ .

$e^{- \frac{2 t}{25} - \frac{2 C}{25}} = | 2.4 - \frac{2 u}{25} |$

Step 3.5

Solve for $u$ .

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Step 3.5.1

Rewrite the equation as $| 2.4 - \frac{2 u}{25} | = e^{- \frac{2 t}{25} - \frac{2 C}{25}}$ .

$| 2.4 - \frac{2 u}{25} | = e^{- \frac{2 t}{25} - \frac{2 C}{25}}$

Step 3.5.2

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

$2.4 - \frac{2 u}{25} = \pm e^{- \frac{2 t}{25} - \frac{2 C}{25}}$

Step 3.5.3

Subtract $2.4$ from both sides of the equation.

$- \frac{2 u}{25} = \pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4$

Step 3.5.4

Multiply both sides of the equation by $- \frac{25}{2}$ .

$- \frac{25}{2} (- \frac{2 u}{25}) = - \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4)$

Step 3.5.5

Simplify both sides of the equation.

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Step 3.5.5.1

Simplify the left side.

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Step 3.5.5.1.1

Simplify $- \frac{25}{2} (- \frac{2 u}{25})$ .

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Step 3.5.5.1.1.1

Cancel the common factor of $25$ .

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Step 3.5.5.1.1.1.1

Move the leading negative in $- \frac{25}{2}$ into the numerator.

$\frac{- 25}{2} (- \frac{2 u}{25}) = - \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4)$

Step 3.5.5.1.1.1.2

Move the leading negative in $- \frac{2 u}{25}$ into the numerator.

$\frac{- 25}{2} \cdot \frac{- 2 u}{25} = - \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4)$

Step 3.5.5.1.1.1.3

Factor $25$ out of $- 25$ .

$\frac{25 (- 1)}{2} \cdot \frac{- 2 u}{25} = - \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4)$

Step 3.5.5.1.1.1.4

Cancel the common factor.

$\frac{25 \cdot - 1}{2} \cdot \frac{- 2 u}{25} = - \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4)$

Step 3.5.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2} (- 2 u) = - \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4)$

Step 3.5.5.1.1.2

Cancel the common factor of $2$ .

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Step 3.5.5.1.1.2.1

Factor $2$ out of $- 2 u$ .

$\frac{- 1}{2} (2 (- u)) = - \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4)$

Step 3.5.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2} (2 (- u)) = - \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4)$

Step 3.5.5.1.1.2.3

Rewrite the expression.

$- - u = - \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4)$

Step 3.5.5.1.1.3

Multiply.

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Step 3.5.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 u = - \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4)$

Step 3.5.5.1.1.3.2

Multiply $u$ by $1$ .

$u = - \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4)$

Step 3.5.5.2

Simplify the right side.

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Step 3.5.5.2.1

Simplify $- \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}} - 2.4)$ .

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Step 3.5.5.2.1.1

Apply the distributive property.

$u = - \frac{25}{2} (\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}}) - \frac{25}{2} \cdot - 2.4$

Step 3.5.5.2.1.2

Combine $\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}}$ and $\frac{25}{2}$ .

$u = - \frac{(\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}}) \cdot 25}{2} - \frac{25}{2} \cdot - 2.4$

Step 3.5.5.2.1.3

Multiply $- \frac{25}{2} \cdot - 2.4$ .

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Step 3.5.5.2.1.3.1

Multiply $- 2.4$ by $- 1$ .

$u = - \frac{(\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}}) \cdot 25}{2} + 2.4 (\frac{25}{2})$

Step 3.5.5.2.1.3.2

Combine $2.4$ and $\frac{25}{2}$ .

$u = - \frac{(\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}}) \cdot 25}{2} + \frac{2.4 \cdot 25}{2}$

Step 3.5.5.2.1.3.3

Multiply $2.4$ by $25$ .

$u = - \frac{(\pm e^{- \frac{2 t}{25} - \frac{2 C}{25}}) \cdot 25}{2} + \frac{60}{2}$

Step 3.5.5.2.1.4

Simplify each term.

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Step 3.5.5.2.1.4.1

Simplify the numerator.

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Step 3.5.5.2.1.4.1.1

Combine the numerators over the common denominator.

$u = - \frac{(\pm e^{\frac{- 2 t - 2 C}{25}}) \cdot 25}{2} + \frac{60}{2}$

Step 3.5.5.2.1.4.1.2

Factor $2$ out of $- 2 t - 2 C$ .

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Step 3.5.5.2.1.4.1.2.1

Factor $2$ out of $- 2 t$ .

$u = - \frac{(\pm e^{\frac{2 (- t) - 2 C}{25}}) \cdot 25}{2} + \frac{60}{2}$

Step 3.5.5.2.1.4.1.2.2

Factor $2$ out of $- 2 C$ .

$u = - \frac{(\pm e^{\frac{2 (- t) + 2 (- C)}{25}}) \cdot 25}{2} + \frac{60}{2}$

Step 3.5.5.2.1.4.1.2.3

Factor $2$ out of $2 (- t) + 2 (- C)$ .

$u = - \frac{(\pm e^{\frac{2 (- t - C)}{25}}) \cdot 25}{2} + \frac{60}{2}$

Step 3.5.5.2.1.4.2

Move $25$ to the left of $\pm e^{\frac{2 (- t - C)}{25}}$ .

$u = - \frac{25 \cdot (\pm e^{\frac{2 (- t - C)}{25}})}{2} + \frac{60}{2}$

Step 3.5.5.2.1.4.3

Divide $60$ by $2$ .

$u = - \frac{25 (\pm e^{\frac{2 (- t - C)}{25}})}{2} + 30$

Step 3.5.5.2.1.5

To write $30$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$u = - \frac{25 (\pm e^{\frac{2 (- t - C)}{25}})}{2} + 30 \cdot \frac{2}{2}$

Step 3.5.5.2.1.6

Simplify terms.

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Step 3.5.5.2.1.6.1

Combine $30$ and $\frac{2}{2}$ .

$u = - \frac{25 (\pm e^{\frac{2 (- t - C)}{25}})}{2} + \frac{30 \cdot 2}{2}$

Step 3.5.5.2.1.6.2

Combine the numerators over the common denominator.

$u = \frac{- 25 (\pm e^{\frac{2 (- t - C)}{25}}) + 30 \cdot 2}{2}$

Step 3.5.5.2.1.6.3

Cancel the common factor of $- 25 (\pm e^{\frac{2 (- t - C)}{25}}) + 30 \cdot 2$ and $2$ .

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Step 3.5.5.2.1.6.3.1

Factor $1$ out of $- 25 (\pm e^{\frac{2 (- t - C)}{25}})$ .

$u = \frac{1 (- 25 (\pm e^{\frac{2 (- t - C)}{25}})) + 30 \cdot 2}{2}$

Step 3.5.5.2.1.6.3.2

Factor $1$ out of $30 \cdot 2$ .

$u = \frac{1 (- 25 (\pm e^{\frac{2 (- t - C)}{25}})) + 1 (30 \cdot 2)}{2}$

Step 3.5.5.2.1.6.3.3

Factor $1$ out of $1 (- 25 (\pm e^{\frac{2 (- t - C)}{25}})) + 1 (30 \cdot 2)$ .

$u = \frac{1 (- 25 (\pm e^{\frac{2 (- t - C)}{25}}) + 30 \cdot 2)}{2}$

Step 3.5.5.2.1.6.3.4

Cancel the common factors.

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Step 3.5.5.2.1.6.3.4.1

Rewrite $2$ as $1 (2)$ .

$u = \frac{1 (- 25 (\pm e^{\frac{2 (- t - C)}{25}}) + 30 \cdot 2)}{1 (2)}$

Step 3.5.5.2.1.6.3.4.2

Cancel the common factor.

$u = \frac{1 (- 25 (\pm e^{\frac{2 (- t - C)}{25}}) + 30 \cdot 2)}{1 \cdot 2}$

Step 3.5.5.2.1.6.3.4.3

Rewrite the expression.

$u = \frac{- 25 (\pm e^{\frac{2 (- t - C)}{25}}) + 30 \cdot 2}{2}$

Step 3.5.5.2.1.7

Simplify the numerator.

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Step 3.5.5.2.1.7.1

Factor $5$ out of $- 25 (\pm e^{\frac{2 (- t - C)}{25}}) + 30 \cdot 2$ .

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Step 3.5.5.2.1.7.1.1

Factor $5$ out of $- 25 (\pm e^{\frac{2 (- t - C)}{25}})$ .

$u = \frac{5 (- 5 (\pm e^{\frac{2 (- t - C)}{25}})) + 30 \cdot 2}{2}$

Step 3.5.5.2.1.7.1.2

Factor $5$ out of $30 \cdot 2$ .

$u = \frac{5 (- 5 (\pm e^{\frac{2 (- t - C)}{25}})) + 5 (6 \cdot 2)}{2}$

Step 3.5.5.2.1.7.1.3

Factor $5$ out of $5 (- 5 (\pm e^{\frac{2 (- t - C)}{25}})) + 5 (6 \cdot 2)$ .

$u = \frac{5 (- 5 (\pm e^{\frac{2 (- t - C)}{25}}) + 6 \cdot 2)}{2}$

Step 3.5.5.2.1.7.2

Multiply $6$ by $2$ .

$u = \frac{5 (- 5 (\pm e^{\frac{2 (- t - C)}{25}}) + 12)}{2}$

Step 3.5.5.2.1.8

Simplify with factoring out.

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Step 3.5.5.2.1.8.1

Factor $- 1$ out of $- 5 (\pm e^{\frac{2 (- t - C)}{25}})$ .

$u = \frac{5 (- (5 (\pm e^{\frac{2 (- t - C)}{25}})) + 12)}{2}$

Step 3.5.5.2.1.8.2

Rewrite $12$ as $- 1 (- 12)$ .

$u = \frac{5 (- (5 (\pm e^{\frac{2 (- t - C)}{25}})) - 1 (- 12))}{2}$

Step 3.5.5.2.1.8.3

Factor $- 1$ out of $- (5 (\pm e^{\frac{2 (- t - C)}{25}})) - 1 (- 12)$ .

$u = \frac{5 (- (5 (\pm e^{\frac{2 (- t - C)}{25}}) - 12))}{2}$

Step 3.5.5.2.1.8.4

Simplify the expression.

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Step 3.5.5.2.1.8.4.1

Rewrite $- (5 (\pm e^{\frac{2 (- t - C)}{25}}) - 12)$ as $- 1 (5 (\pm e^{\frac{2 (- t - C)}{25}}) - 12)$ .

$u = \frac{5 (- 1 (5 (\pm e^{\frac{2 (- t - C)}{25}}) - 12))}{2}$

Step 3.5.5.2.1.8.4.2

Move the negative in front of the fraction.

$u = - \frac{5 (5 (\pm e^{\frac{2 (- t - C)}{25}}) - 12)}{2}$

Step 4

Simplify the constant of integration.

$u = - \frac{5 (5 (\pm e^{\frac{2 (- t + C)}{25}}) - 12)}{2}$