Calculus Examples

$\frac{d r}{d θ} = \frac{r^{2}}{θ}$ , $r (1) = 2$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{r^{2}}$ .

$\frac{1}{r^{2}} \frac{d r}{d θ} = \frac{1}{r^{2}} \cdot \frac{r^{2}}{θ}$

Step 1.2

Simplify.

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Step 1.2.1

Combine.

$\frac{1}{r^{2}} \frac{d r}{d θ} = \frac{1 r^{2}}{r^{2} θ}$

Step 1.2.2

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

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Step 1.2.2.1

Cancel the common factor.

$\frac{1}{r^{2}} \frac{d r}{d θ} = \frac{1 r^{2}}{r^{2} θ}$

Step 1.2.2.2

Rewrite the expression.

$\frac{1}{r^{2}} \frac{d r}{d θ} = \frac{1}{θ}$

Step 1.3

Rewrite the equation.

$\frac{1}{r^{2}} d r = \frac{1}{θ} d θ$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{r^{2}} d r = \int \frac{1}{θ} d θ$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

Move $r^{2}$ out of the denominator by raising it to the $- 1$ power.

$\int {(r^{2})}^{- 1} d r = \int \frac{1}{θ} d θ$

Step 2.2.1.2

Multiply the exponents in ${(r^{2})}^{- 1}$ .

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Step 2.2.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int r^{2 \cdot - 1} d r = \int \frac{1}{θ} d θ$

Step 2.2.1.2.2

Multiply $2$ by $- 1$ .

$\int r^{- 2} d r = \int \frac{1}{θ} d θ$

Step 2.2.2

By the Power Rule, the integral of $r^{- 2}$ with respect to $r$ is $- r^{- 1}$ .

$- r^{- 1} + C_{1} = \int \frac{1}{θ} d θ$

Step 2.2.3

Rewrite $- r^{- 1} + C_{1}$ as $- \frac{1}{r} + C_{1}$ .

$- \frac{1}{r} + C_{1} = \int \frac{1}{θ} d θ$

Step 2.3

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

$- \frac{1}{r} + C_{1} = \ln (| θ |) + C_{2}$

Step 2.4

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

$- \frac{1}{r} = \ln (| θ |) + C$

Step 3

Solve for $r$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$r, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$r$

Step 3.2

Multiply each term in $- \frac{1}{r} = \ln (| θ |) + C$ by $r$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $- \frac{1}{r} = \ln (| θ |) + C$ by $r$ .

$- \frac{1}{r} r = \ln (| θ |) r + C r$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $r$ .

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Step 3.2.2.1.1

Move the leading negative in $- \frac{1}{r}$ into the numerator.

$\frac{- 1}{r} r = \ln (| θ |) r + C r$

Step 3.2.2.1.2

Cancel the common factor.

$\frac{- 1}{r} r = \ln (| θ |) r + C r$

Step 3.2.2.1.3

Rewrite the expression.

$- 1 = \ln (| θ |) r + C r$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Reorder factors in $\ln (| θ |) r + C r$ .

$- 1 = r \ln (| θ |) + C r$

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $r \ln (| θ |) + C r = - 1$ .

$r \ln (| θ |) + C r = - 1$

Step 3.3.2

Factor $r$ out of $r \ln (| θ |) + C r$ .

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Step 3.3.2.1

Factor $r$ out of $C r$ .

$r \ln (| θ |) + r C = - 1$

Step 3.3.2.2

Factor $r$ out of $r \ln (| θ |) + r C$ .

$r (\ln (| θ |) + C) = - 1$

Step 3.3.3

Divide each term in $r (\ln (| θ |) + C) = - 1$ by $\ln (| θ |) + C$ and simplify.

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Step 3.3.3.1

Divide each term in $r (\ln (| θ |) + C) = - 1$ by $\ln (| θ |) + C$ .

$\frac{r (\ln (| θ |) + C)}{\ln (| θ |) + C} = \frac{- 1}{\ln (| θ |) + C}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $\ln (| θ |) + C$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{r (\ln (| θ |) + C)}{\ln (| θ |) + C} = \frac{- 1}{\ln (| θ |) + C}$

Step 3.3.3.2.1.2

Divide $r$ by $1$ .

$r = \frac{- 1}{\ln (| θ |) + C}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Move the negative in front of the fraction.

$r = - \frac{1}{\ln (| θ |) + C}$

Step 4

Use the initial condition to find the value of $C$ by substituting $1$ for $θ$ and $2$ for $r$ in $r = - \frac{1}{\ln (| θ |) + C}$ .

$2 = - \frac{1}{\ln (| 1 |) + C}$

Step 5

Solve for $C$ .

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Step 5.1

Rewrite the equation as $- \frac{1}{\ln (| 1 |) + C} = 2$ .

$- \frac{1}{\ln (| 1 |) + C} = 2$

Step 5.2

Factor each term.

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Step 5.2.1

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$- \frac{1}{\ln (1) + C} = 2$

Step 5.2.2

The natural logarithm of $1$ is $0$ .

$- \frac{1}{0 + C} = 2$

Step 5.2.3

Add $0$ and $C$ .

$- \frac{1}{C} = 2$

Step 5.3

Find the LCD of the terms in the equation.

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Step 5.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$C, 1$

Step 5.3.2

The LCM of one and any expression is the expression.

$C$

Step 5.4

Multiply each term in $- \frac{1}{C} = 2$ by $C$ to eliminate the fractions.

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Step 5.4.1

Multiply each term in $- \frac{1}{C} = 2$ by $C$ .

$- \frac{1}{C} C = 2 C$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $C$ .

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Step 5.4.2.1.1

Move the leading negative in $- \frac{1}{C}$ into the numerator.

$\frac{- 1}{C} C = 2 C$

Step 5.4.2.1.2

Cancel the common factor.

$\frac{- 1}{C} C = 2 C$

Step 5.4.2.1.3

Rewrite the expression.

$- 1 = 2 C$

Step 5.5

Solve the equation.

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Step 5.5.1

Rewrite the equation as $2 C = - 1$ .

$2 C = - 1$

Step 5.5.2

Divide each term in $2 C = - 1$ by $2$ and simplify.

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Step 5.5.2.1

Divide each term in $2 C = - 1$ by $2$ .

$\frac{2 C}{2} = \frac{- 1}{2}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Cancel the common factor of $2$ .

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Step 5.5.2.2.1.1

Cancel the common factor.

$\frac{2 C}{2} = \frac{- 1}{2}$

Step 5.5.2.2.1.2

Divide $C$ by $1$ .

$C = \frac{- 1}{2}$

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Move the negative in front of the fraction.

$C = - \frac{1}{2}$

Step 6

Substitute $- \frac{1}{2}$ for $C$ in $r = - \frac{1}{\ln (| θ |) + C}$ and simplify.

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Step 6.1

Substitute $- \frac{1}{2}$ for $C$ .

$r = - \frac{1}{\ln (| θ |) - \frac{1}{2}}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

To write $\ln (| θ |)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$r = - \frac{1}{\ln (| θ |) \cdot \frac{2}{2} - \frac{1}{2}}$

Step 6.2.2

Combine $\ln (| θ |)$ and $\frac{2}{2}$ .

$r = - \frac{1}{\frac{\ln (| θ |) \cdot 2}{2} - \frac{1}{2}}$

Step 6.2.3

Combine the numerators over the common denominator.

$r = - \frac{1}{\frac{\ln (| θ |) \cdot 2 - 1}{2}}$

Step 6.2.4

Simplify the numerator.

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Step 6.2.4.1

Multiply $\ln (| θ |) \cdot 2$ .

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Step 6.2.4.1.1

Reorder $\ln (| θ |)$ and $2$ .

$r = - \frac{1}{\frac{2 \cdot \ln (| θ |) - 1}{2}}$

Step 6.2.4.1.2

Simplify $2 \cdot \ln (| θ |)$ by moving $2$ inside the logarithm.

$r = - \frac{1}{\frac{\ln ({| θ |}^{2}) - 1}{2}}$

Step 6.2.4.2

Remove the absolute value in ${| θ |}^{2}$ because exponentiations with even powers are always positive.

$r = - \frac{1}{\frac{\ln (θ^{2}) - 1}{2}}$

Step 6.3

Multiply the numerator by the reciprocal of the denominator.

$r = - (1 \frac{2}{\ln (θ^{2}) - 1})$

Step 6.4

Multiply $\frac{2}{\ln (θ^{2}) - 1}$ by $1$ .

$r = - \frac{2}{\ln (θ^{2}) - 1}$