Calculus Examples

$\frac{d r}{d t} = \frac{2 t + t r^{2}}{4 r + r t^{2}}$

Step 1

Separate the variables.

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Step 1.1

Factor $t$ out of $2 t + t r^{2}$ .

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Step 1.1.1

Factor $t$ out of $2 t$ .

$\frac{d r}{d t} = \frac{t \cdot 2 + t r^{2}}{4 r + r t^{2}}$

Step 1.1.2

Factor $t$ out of $t r^{2}$ .

$\frac{d r}{d t} = \frac{t \cdot 2 + t (r^{2})}{4 r + r t^{2}}$

Step 1.1.3

Factor $t$ out of $t \cdot 2 + t (r^{2})$ .

$\frac{d r}{d t} = \frac{t \cdot (2 + r^{2})}{4 r + r t^{2}}$

Step 1.1.4

Multiply $t$ by $2 + r^{2}$ .

$\frac{d r}{d t} = \frac{t (2 + r^{2})}{4 r + r t^{2}}$

Step 1.2

Factor $r$ out of $4 r + r t^{2}$ .

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Step 1.2.1

Factor $r$ out of $4 r$ .

$\frac{d r}{d t} = \frac{t (2 + r^{2})}{r \cdot 4 + r t^{2}}$

Step 1.2.2

Factor $r$ out of $r t^{2}$ .

$\frac{d r}{d t} = \frac{t (2 + r^{2})}{r \cdot 4 + r (t^{2})}$

Step 1.2.3

Factor $r$ out of $r \cdot 4 + r (t^{2})$ .

$\frac{d r}{d t} = \frac{t (2 + r^{2})}{r \cdot (4 + t^{2})}$

Step 1.2.4

Multiply $r$ by $4 + t^{2}$ .

$\frac{d r}{d t} = \frac{t (2 + r^{2})}{r (4 + t^{2})}$

Step 1.3

Regroup factors.

$\frac{d r}{d t} = \frac{t}{4 + t^{2}} \cdot \frac{2 + r^{2}}{r}$

Step 1.4

Multiply both sides by $\frac{r}{2 + r^{2}}$ .

$\frac{r}{2 + r^{2}} \frac{d r}{d t} = \frac{r}{2 + r^{2}} (\frac{t}{4 + t^{2}} \cdot \frac{2 + r^{2}}{r})$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $\frac{t}{4 + t^{2}}$ by $\frac{2 + r^{2}}{r}$ .

$\frac{r}{2 + r^{2}} \frac{d r}{d t} = \frac{r}{2 + r^{2}} \cdot \frac{t (2 + r^{2})}{(4 + t^{2}) r}$

Step 1.5.2

Cancel the common factor of $r$ .

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Step 1.5.2.1

Factor $r$ out of $(4 + t^{2}) r$ .

$\frac{r}{2 + r^{2}} \frac{d r}{d t} = \frac{r}{2 + r^{2}} \cdot \frac{t (2 + r^{2})}{r (4 + t^{2})}$

Step 1.5.2.2

Cancel the common factor.

$\frac{r}{2 + r^{2}} \frac{d r}{d t} = \frac{r}{2 + r^{2}} \cdot \frac{t (2 + r^{2})}{r (4 + t^{2})}$

Step 1.5.2.3

Rewrite the expression.

$\frac{r}{2 + r^{2}} \frac{d r}{d t} = \frac{1}{2 + r^{2}} \cdot \frac{t (2 + r^{2})}{4 + t^{2}}$

Step 1.5.3

Cancel the common factor of $2 + r^{2}$ .

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Step 1.5.3.1

Factor $2 + r^{2}$ out of $t (2 + r^{2})$ .

$\frac{r}{2 + r^{2}} \frac{d r}{d t} = \frac{1}{2 + r^{2}} \cdot \frac{(2 + r^{2}) t}{4 + t^{2}}$

Step 1.5.3.2

Cancel the common factor.

$\frac{r}{2 + r^{2}} \frac{d r}{d t} = \frac{1}{2 + r^{2}} \cdot \frac{(2 + r^{2}) t}{4 + t^{2}}$

Step 1.5.3.3

Rewrite the expression.

$\frac{r}{2 + r^{2}} \frac{d r}{d t} = \frac{t}{4 + t^{2}}$

Step 1.6

Rewrite the equation.

$\frac{r}{2 + r^{2}} d r = \frac{t}{4 + t^{2}} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{r}{2 + r^{2}} d r = \int \frac{t}{4 + t^{2}} d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u_{1} = 2 + r^{2}$ . Then $d u_{1} = 2 r d r$ , so $\frac{1}{2} d u_{1} = r d r$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.2.1.1

Let $u_{1} = 2 + r^{2}$ . Find $\frac{d u_{1}}{d r}$ .

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Step 2.2.1.1.1

Differentiate $2 + r^{2}$ .

$\frac{d}{d r} [2 + r^{2}]$

Step 2.2.1.1.2

By the Sum Rule, the derivative of $2 + r^{2}$ with respect to $r$ is $\frac{d}{d r} [2] + \frac{d}{d r} [r^{2}]$ .

$\frac{d}{d r} [2] + \frac{d}{d r} [r^{2}]$

Step 2.2.1.1.3

Since $2$ is constant with respect to $r$ , the derivative of $2$ with respect to $r$ is $0$ .

$0 + \frac{d}{d r} [r^{2}]$

Step 2.2.1.1.4

Differentiate using the Power Rule which states that $\frac{d}{d r} [r^{n}]$ is $n r^{n - 1}$ where $n = 2$ .

$0 + 2 r$

Step 2.2.1.1.5

Add $0$ and $2 r$ .

$2 r$

Step 2.2.1.2

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

$\int \frac{1}{u_{1}} \cdot \frac{1}{2} d u_{1} = \int \frac{t}{4 + t^{2}} d t$

Step 2.2.2

Simplify.

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Step 2.2.2.1

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

$\int \frac{1}{u_{1} \cdot 2} d u_{1} = \int \frac{t}{4 + t^{2}} d t$

Step 2.2.2.2

Move $2$ to the left of $u_{1}$ .

$\int \frac{1}{2 u_{1}} d u_{1} = \int \frac{t}{4 + t^{2}} d t$

Step 2.2.3

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

$\frac{1}{2} \int \frac{1}{u_{1}} d u_{1} = \int \frac{t}{4 + t^{2}} d t$

Step 2.2.4

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

$\frac{1}{2} (\ln (| u_{1} |) + C_{1}) = \int \frac{t}{4 + t^{2}} d t$

Step 2.2.5

Simplify.

$\frac{1}{2} \ln (| u_{1} |) + C_{1} = \int \frac{t}{4 + t^{2}} d t$

Step 2.2.6

Replace all occurrences of $u_{1}$ with $2 + r^{2}$ .

$\frac{1}{2} \ln (| 2 + r^{2} |) + C_{1} = \int \frac{t}{4 + t^{2}} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u_{2} = 4 + t^{2}$ . Then $d u_{2} = 2 t d t$ , so $\frac{1}{2} d u_{2} = t d t$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.1.1

Let $u_{2} = 4 + t^{2}$ . Find $\frac{d u_{2}}{d t}$ .

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Step 2.3.1.1.1

Differentiate $4 + t^{2}$ .

$\frac{d}{d t} [4 + t^{2}]$

Step 2.3.1.1.2

By the Sum Rule, the derivative of $4 + t^{2}$ with respect to $t$ is $\frac{d}{d t} [4] + \frac{d}{d t} [t^{2}]$ .

$\frac{d}{d t} [4] + \frac{d}{d t} [t^{2}]$

Step 2.3.1.1.3

Since $4$ is constant with respect to $t$ , the derivative of $4$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [t^{2}]$

Step 2.3.1.1.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$0 + 2 t$

Step 2.3.1.1.5

Add $0$ and $2 t$ .

$2 t$

Step 2.3.1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{2} \ln (| 2 + r^{2} |) + C_{1} = \int \frac{1}{u_{2}} \cdot \frac{1}{2} d u_{2}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

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

$\frac{1}{2} \ln (| 2 + r^{2} |) + C_{1} = \int \frac{1}{u_{2} \cdot 2} d u_{2}$

Step 2.3.2.2

Move $2$ to the left of $u_{2}$ .

$\frac{1}{2} \ln (| 2 + r^{2} |) + C_{1} = \int \frac{1}{2 u_{2}} d u_{2}$

Step 2.3.3

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

$\frac{1}{2} \ln (| 2 + r^{2} |) + C_{1} = \frac{1}{2} \int \frac{1}{u_{2}} d u_{2}$

Step 2.3.4

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\frac{1}{2} \ln (| 2 + r^{2} |) + C_{1} = \frac{1}{2} (\ln (| u_{2} |) + C_{2})$

Step 2.3.5

Simplify.

$\frac{1}{2} \ln (| 2 + r^{2} |) + C_{1} = \frac{1}{2} \ln (| u_{2} |) + C_{2}$

Step 2.3.6

Replace all occurrences of $u_{2}$ with $4 + t^{2}$ .

$\frac{1}{2} \ln (| 2 + r^{2} |) + C_{1} = \frac{1}{2} \ln (| 4 + t^{2} |) + C_{2}$

Step 2.4

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

$\frac{1}{2} \ln (| 2 + r^{2} |) = \frac{1}{2} \ln (| 4 + t^{2} |) + C$

Step 3

Solve for $r$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \ln (| 2 + r^{2} |)) = 2 (\frac{1}{2} \ln (| 4 + t^{2} |) + C)$

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 $2 (\frac{1}{2} \ln (| 2 + r^{2} |))$ .

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Step 3.2.1.1.1

Combine $\frac{1}{2}$ and $\ln (| 2 + r^{2} |)$ .

$2 \frac{\ln (| 2 + r^{2} |)}{2} = 2 (\frac{1}{2} \ln (| 4 + t^{2} |) + C)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{\ln (| 2 + r^{2} |)}{2} = 2 (\frac{1}{2} \ln (| 4 + t^{2} |) + C)$

Step 3.2.1.1.2.2

Rewrite the expression.

$\ln (| 2 + r^{2} |) = 2 (\frac{1}{2} \ln (| 4 + t^{2} |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{1}{2} \ln (| 4 + t^{2} |) + C)$ .

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Step 3.2.2.1.1

Combine $\frac{1}{2}$ and $\ln (| 4 + t^{2} |)$ .

$\ln (| 2 + r^{2} |) = 2 (\frac{\ln (| 4 + t^{2} |)}{2} + C)$

Step 3.2.2.1.2

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

$\ln (| 2 + r^{2} |) = 2 (\frac{\ln (| 4 + t^{2} |)}{2} + C \cdot \frac{2}{2})$

Step 3.2.2.1.3

Simplify terms.

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Step 3.2.2.1.3.1

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

$\ln (| 2 + r^{2} |) = 2 (\frac{\ln (| 4 + t^{2} |)}{2} + \frac{C \cdot 2}{2})$

Step 3.2.2.1.3.2

Combine the numerators over the common denominator.

$\ln (| 2 + r^{2} |) = 2 \frac{\ln (| 4 + t^{2} |) + C \cdot 2}{2}$

Step 3.2.2.1.3.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.3.1

Cancel the common factor.

$\ln (| 2 + r^{2} |) = 2 \frac{\ln (| 4 + t^{2} |) + C \cdot 2}{2}$

Step 3.2.2.1.3.3.2

Rewrite the expression.

$\ln (| 2 + r^{2} |) = \ln (| 4 + t^{2} |) + C \cdot 2$

Step 3.2.2.1.4

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

$\ln (| 2 + r^{2} |) = \ln (| 4 + t^{2} |) + 2 C$

Step 3.3

Move all the terms containing a logarithm to the left side of the equation.

$\ln (| 2 + r^{2} |) - \ln (| 4 + t^{2} |) = 2 C$

Step 3.4

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| 2 + r^{2} |}{| 4 + t^{2} |}) = 2 C$

Step 3.5

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

$e^{\ln (\frac{| 2 + r^{2} |}{| 4 + t^{2} |})} = e^{2 C}$

Step 3.6

Rewrite $\ln (\frac{| 2 + r^{2} |}{| 4 + t^{2} |}) = 2 C$ 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 $b^{y} = x$ .

$e^{2 C} = \frac{| 2 + r^{2} |}{| 4 + t^{2} |}$

Step 3.7

Solve for $r$ .

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Step 3.7.1

Rewrite the equation as $\frac{| 2 + r^{2} |}{| 4 + t^{2} |} = e^{2 C}$ .

$\frac{| 2 + r^{2} |}{| 4 + t^{2} |} = e^{2 C}$

Step 3.7.2

Multiply both sides by $| 4 + t^{2} |$ .

$\frac{| 2 + r^{2} |}{| 4 + t^{2} |} | 4 + t^{2} | = e^{2 C} | 4 + t^{2} |$

Step 3.7.3

Simplify the left side.

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Step 3.7.3.1

Cancel the common factor of $| 4 + t^{2} |$ .

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Step 3.7.3.1.1

Cancel the common factor.

$\frac{| 2 + r^{2} |}{| 4 + t^{2} |} | 4 + t^{2} | = e^{2 C} | 4 + t^{2} |$

Step 3.7.3.1.2

Rewrite the expression.

$| 2 + r^{2} | = e^{2 C} | 4 + t^{2} |$

Step 3.7.4

Solve for $r$ .

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Step 3.7.4.1

Reorder factors in $e^{2 C} | 4 + t^{2} |$ .

$| 2 + r^{2} | = | 4 + t^{2} | e^{2 C}$

Step 3.7.4.2

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

$2 + r^{2} = \pm | 4 + t^{2} | e^{2 C}$

Step 3.7.4.3

Reorder factors in $\pm | 4 + t^{2} | e^{2 C}$ .

$2 + r^{2} = \pm e^{2 C} | 4 + t^{2} |$

Step 3.7.4.4

Subtract $2$ from both sides of the equation.

$r^{2} = \pm e^{2 C} | 4 + t^{2} | - 2$

Step 3.7.4.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$r = \pm \sqrt{\pm e^{2 C} | 4 + t^{2} | - 2}$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$r = \pm \sqrt{\pm C | 4 + t^{2} | - 2}$

Step 4.2

Combine constants with the plus or minus.

$r = \pm \sqrt{C | 4 + t^{2} | - 2}$