Calculus Examples

$\frac{d r}{d θ} = - r \tan (θ)$ , $r (π) = 2$

Step 1

Separate the variables.

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Step 1.1

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

$\frac{1}{r} \frac{d r}{d θ} = \frac{1}{r} (- r \tan (θ))$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{r} \frac{d r}{d θ} = - \frac{1}{r} (r \tan (θ))$

Step 1.2.2

Cancel the common factor of $r$ .

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Step 1.2.2.1

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

$\frac{1}{r} \frac{d r}{d θ} = \frac{- 1}{r} (r \tan (θ))$

Step 1.2.2.2

Factor $r$ out of $r \tan (θ)$ .

$\frac{1}{r} \frac{d r}{d θ} = \frac{- 1}{r} (r (\tan (θ)))$

Step 1.2.2.3

Cancel the common factor.

$\frac{1}{r} \frac{d r}{d θ} = \frac{- 1}{r} (r \tan (θ))$

Step 1.2.2.4

Rewrite the expression.

$\frac{1}{r} \frac{d r}{d θ} = - \tan (θ)$

Step 1.3

Rewrite the equation.

$\frac{1}{r} d r = - \tan (θ) d θ$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

$\ln (| r |) + C_{1} = \int - \tan (θ) d θ$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$\ln (| r |) + C_{1} = - \int \tan (θ) d θ$

Step 2.3.2

The integral of $\tan (θ)$ with respect to $θ$ is $\ln (| \sec (θ) |)$ .

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

Step 2.3.3

Simplify.

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

Step 2.4

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

$\ln (| r |) = - \ln (| \sec (θ) |) + C$

Step 3

Solve for $r$ .

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Step 3.1

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

$\ln (| r |) + \ln (| \sec (θ) |) = C$

Step 3.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (| r | | \sec (θ) |) = C$

Step 3.3

To multiply absolute values, multiply the terms inside each absolute value.

$\ln (| r \sec (θ) |) = C$

Step 3.4

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

$e^{\ln (| r \sec (θ) |)} = e^{C}$

Step 3.5

Rewrite $\ln (| r \sec (θ) |) = C$ 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 $b^{y} = x$ .

$e^{C} = | r \sec (θ) |$

Step 3.6

Solve for $r$ .

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Step 3.6.1

Rewrite the equation as $| r \sec (θ) | = e^{C}$ .

$| r \sec (θ) | = e^{C}$

Step 3.6.2

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

$r \sec (θ) = \pm e^{C}$

Step 3.6.3

Divide each term in $r \sec (θ) = \pm e^{C}$ by $\sec (θ)$ and simplify.

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Step 3.6.3.1

Divide each term in $r \sec (θ) = \pm e^{C}$ by $\sec (θ)$ .

$\frac{r \sec (θ)}{\sec (θ)} = \frac{\pm e^{C}}{\sec (θ)}$

Step 3.6.3.2

Simplify the left side.

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Step 3.6.3.2.1

Cancel the common factor of $\sec (θ)$ .

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Step 3.6.3.2.1.1

Cancel the common factor.

$\frac{r \sec (θ)}{\sec (θ)} = \frac{\pm e^{C}}{\sec (θ)}$

Step 3.6.3.2.1.2

Divide $r$ by $1$ .

$r = \frac{\pm e^{C}}{\sec (θ)}$

Step 3.6.3.3

Simplify the right side.

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Step 3.6.3.3.1

Separate fractions.

$r = \frac{\pm e^{C}}{1} \cdot \frac{1}{\sec (θ)}$

Step 3.6.3.3.2

Rewrite $\sec (θ)$ in terms of sines and cosines.

$r = \frac{\pm e^{C}}{1} \cdot \frac{1}{\frac{1}{\cos (θ)}}$

Step 3.6.3.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (θ)}$ .

$r = \frac{\pm e^{C}}{1} (1 \cos (θ))$

Step 3.6.3.3.4

Multiply $\cos (θ)$ by $1$ .

$r = \frac{\pm e^{C}}{1} \cos (θ)$

Step 3.6.3.3.5

Divide $\pm e^{C}$ by $1$ .

$r = (\pm e^{C}) \cos (θ)$

Step 3.6.3.3.6

Reorder factors in $(\pm e^{C}) \cos (θ)$ .

$r = \cos (θ) (\pm e^{C})$

Step 4

Simplify the constant of integration.

$r = \cos (θ) C$

Step 5

Use the initial condition to find the value of $C$ by substituting $π$ for $θ$ and $2$ for $r$ in $r = \cos (θ) C$ .

$2 = \cos (π) C$

Step 6

Solve for $C$ .

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Step 6.1

Rewrite the equation as $\cos (π) C = 2$ .

$\cos (π) C = 2$

Step 6.2

Divide each term in $\cos (π) C = 2$ by $\cos (π)$ and simplify.

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Step 6.2.1

Divide each term in $\cos (π) C = 2$ by $\cos (π)$ .

$\frac{\cos (π) C}{\cos (π)} = \frac{2}{\cos (π)}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $\cos (π)$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{\cos (π) C}{\cos (π)} = \frac{2}{\cos (π)}$

Step 6.2.2.1.2

Divide $C$ by $1$ .

$C = \frac{2}{\cos (π)}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Separate fractions.

$C = \frac{2}{1} \cdot \frac{1}{\cos (π)}$

Step 6.2.3.2

Convert from $\frac{1}{\cos (π)}$ to $\sec (π)$ .

$C = \frac{2}{1} \sec (π)$

Step 6.2.3.3

Divide $2$ by $1$ .

$C = 2 \sec (π)$

Step 6.2.3.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because secant is negative in the second quadrant.

$C = 2 (- \sec (0))$

Step 6.2.3.5

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

$C = 2 (- 1 \cdot 1)$

Step 6.2.3.6

Multiply $2 (- 1 \cdot 1)$ .

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Step 6.2.3.6.1

Multiply $- 1$ by $1$ .

$C = 2 \cdot - 1$

Step 6.2.3.6.2

Multiply $2$ by $- 1$ .

$C = - 2$

Step 7

Substitute $- 2$ for $C$ in $r = \cos (θ) C$ and simplify.

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Step 7.1

Substitute $- 2$ for $C$ .

$r = \cos (θ) \cdot - 2$

Step 7.2

Move $- 2$ to the left of $\cos (θ)$ .

$r = - 2 \cos (θ)$