Calculus Examples

$\frac{d r}{d θ} = \csc^{2} (7 θ) \cot (7 θ)$ , $r (\frac{π}{4}) = 3$

Step 1

Rewrite the equation.

$d r = \csc^{2} (7 θ) \cot (7 θ) d θ$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d r = \int \csc^{2} (7 θ) \cot (7 θ) d θ$

Step 2.2

Apply the constant rule.

$r + C_{1} = \int \csc^{2} (7 θ) \cot (7 θ) d θ$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u_{2} = \csc (7 θ)$ . Then $d u_{2} = - 7 \cot (7 θ) \csc (7 θ) d θ$ , so $- \frac{1}{7} d u_{2} = \cot (7 θ) \csc (7 θ) d θ$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.1.1

Let $u_{2} = \csc (7 θ)$ . Find $\frac{d u_{2}}{d θ}$ .

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Step 2.3.1.1.1

Differentiate $\csc (7 θ)$ .

$\frac{d}{d θ} [\csc (7 θ)]$

Step 2.3.1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d θ} [f (g (θ))]$ is $f' (g (θ)) g' (θ)$ where $f (θ) = \csc (θ)$ and $g (θ) = 7 θ$ .

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Step 2.3.1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $7 θ$ .

$\frac{d}{d u_{1}} [\csc (u_{1})] \frac{d}{d θ} [7 θ]$

Step 2.3.1.1.2.2

The derivative of $\csc (u_{1})$ with respect to $u_{1}$ is $- \csc (u_{1}) \cot (u_{1})$ .

$- \csc (u_{1}) \cot (u_{1}) \frac{d}{d θ} [7 θ]$

Step 2.3.1.1.2.3

Replace all occurrences of $u_{1}$ with $7 θ$ .

$- \csc (7 θ) \cot (7 θ) \frac{d}{d θ} [7 θ]$

Step 2.3.1.1.3

Differentiate.

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Step 2.3.1.1.3.1

Since $7$ is constant with respect to $θ$ , the derivative of $7 θ$ with respect to $θ$ is $7 \frac{d}{d θ} [θ]$ .

$- \csc (7 θ) \cot (7 θ) (7 \frac{d}{d θ} [θ])$

Step 2.3.1.1.3.2

Multiply $7$ by $- 1$ .

$- 7 \csc (7 θ) \cot (7 θ) \frac{d}{d θ} [θ]$

Step 2.3.1.1.3.3

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

$- 7 \csc (7 θ) \cot (7 θ) \cdot 1$

Step 2.3.1.1.3.4

Simplify the expression.

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Step 2.3.1.1.3.4.1

Multiply $- 7$ by $1$ .

$- 7 \csc (7 θ) \cot (7 θ)$

Step 2.3.1.1.3.4.2

Reorder the factors of $- 7 \csc (7 θ) \cot (7 θ)$ .

$- 7 \cot (7 θ) \csc (7 θ)$

Step 2.3.1.2

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

$r + C_{1} = \int u_{2} \frac{1}{- 7} d u_{2}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Move the negative in front of the fraction.

$r + C_{1} = \int u_{2} (- \frac{1}{7}) d u_{2}$

Step 2.3.2.2

Combine $u_{2}$ and $\frac{1}{7}$ .

$r + C_{1} = \int - \frac{u_{2}}{7} d u_{2}$

Step 2.3.3

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

$r + C_{1} = - \int \frac{u_{2}}{7} d u_{2}$

Step 2.3.4

Since $\frac{1}{7}$ is constant with respect to $u_{2}$ , move $\frac{1}{7}$ out of the integral.

$r + C_{1} = - (\frac{1}{7} \int u_{2} d u_{2})$

Step 2.3.5

By the Power Rule, the integral of $u_{2}$ with respect to $u_{2}$ is $\frac{1}{2} {u_{2}}^{2}$ .

$r + C_{1} = - \frac{1}{7} (\frac{1}{2} {u_{2}}^{2} + C_{2})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Rewrite $- \frac{1}{7} (\frac{1}{2} {u_{2}}^{2} + C_{2})$ as $- \frac{1}{7} \cdot \frac{1}{2} {u_{2}}^{2} + C_{2}$ .

$r + C_{1} = - \frac{1}{7} \cdot \frac{1}{2} {u_{2}}^{2} + C_{2}$

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{7}$ .

$r + C_{1} = - \frac{1}{2 \cdot 7} {u_{2}}^{2} + C_{2}$

Step 2.3.6.2.2

Multiply $2$ by $7$ .

$r + C_{1} = - \frac{1}{14} {u_{2}}^{2} + C_{2}$

Step 2.3.7

Replace all occurrences of $u_{2}$ with $\csc (7 θ)$ .

$r + C_{1} = - \frac{1}{14} \csc^{2} (7 θ) + C_{2}$

Step 2.4

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

$r = - \frac{1}{14} \csc^{2} (7 θ) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $\frac{π}{4}$ for $θ$ and $3$ for $r$ in $r = - \frac{1}{14} \csc^{2} (7 θ) + K$ .

$3 = - \frac{1}{14} \csc^{2} (7 \frac{π}{4}) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- \frac{1}{14} \cdot \csc^{2} (7 (\frac{π}{4})) + K = 3$ .

$- \frac{1}{14} \cdot \csc^{2} (7 (\frac{π}{4})) + K = 3$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Combine $7$ and $\frac{π}{4}$ .

$- \frac{1}{14} \cdot \csc^{2} (\frac{7 π}{4}) + K = 3$

Step 4.2.1.2

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

$- \frac{1}{14} \cdot {(- \csc (\frac{π}{4}))}^{2} + K = 3$

Step 4.2.1.3

The exact value of $\csc (\frac{π}{4})$ is $\sqrt{2}$ .

$- \frac{1}{14} \cdot {(- \sqrt{2})}^{2} + K = 3$

Step 4.2.1.4

Apply the product rule to $- \sqrt{2}$ .

$- \frac{1}{14} \cdot ({(- 1)}^{2} {\sqrt{2}}^{2}) + K = 3$

Step 4.2.1.5

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 4.2.1.5.1

Move ${(- 1)}^{2}$ .

${(- 1)}^{2} \cdot - 1 \frac{1}{14} \cdot {\sqrt{2}}^{2} + K = 3$

Step 4.2.1.5.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 4.2.1.5.2.1

Raise $- 1$ to the power of $1$ .

${(- 1)}^{2} \cdot {(- 1)}^{1} \frac{1}{14} \cdot {\sqrt{2}}^{2} + K = 3$

Step 4.2.1.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(- 1)}^{2 + 1} \frac{1}{14} \cdot {\sqrt{2}}^{2} + K = 3$

Step 4.2.1.5.3

Add $2$ and $1$ .

${(- 1)}^{3} \frac{1}{14} \cdot {\sqrt{2}}^{2} + K = 3$

Step 4.2.1.6

Raise $- 1$ to the power of $3$ .

$- \frac{1}{14} \cdot {\sqrt{2}}^{2} + K = 3$

Step 4.2.1.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.2.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$- \frac{1}{14} \cdot {(2^{\frac{1}{2}})}^{2} + K = 3$

Step 4.2.1.7.2

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

$- \frac{1}{14} \cdot 2^{\frac{1}{2} \cdot 2} + K = 3$

Step 4.2.1.7.3

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

$- \frac{1}{14} \cdot 2^{\frac{2}{2}} + K = 3$

Step 4.2.1.7.4

Cancel the common factor of $2$ .

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Step 4.2.1.7.4.1

Cancel the common factor.

$- \frac{1}{14} \cdot 2^{\frac{2}{2}} + K = 3$

Step 4.2.1.7.4.2

Rewrite the expression.

$- \frac{1}{14} \cdot 2^{1} + K = 3$

Step 4.2.1.7.5

Evaluate the exponent.

$- \frac{1}{14} \cdot 2 + K = 3$

Step 4.2.1.8

Cancel the common factor of $2$ .

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Step 4.2.1.8.1

Move the leading negative in $- \frac{1}{14}$ into the numerator.

$\frac{- 1}{14} \cdot 2 + K = 3$

Step 4.2.1.8.2

Factor $2$ out of $14$ .

$\frac{- 1}{2 (7)} \cdot 2 + K = 3$

Step 4.2.1.8.3

Cancel the common factor.

$\frac{- 1}{2 \cdot 7} \cdot 2 + K = 3$

Step 4.2.1.8.4

Rewrite the expression.

$\frac{- 1}{7} + K = 3$

Step 4.2.1.9

Move the negative in front of the fraction.

$- \frac{1}{7} + K = 3$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

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Step 4.3.1

Add $\frac{1}{7}$ to both sides of the equation.

$K = 3 + \frac{1}{7}$

Step 4.3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$K = 3 \cdot \frac{7}{7} + \frac{1}{7}$

Step 4.3.3

Combine $3$ and $\frac{7}{7}$ .

$K = \frac{3 \cdot 7}{7} + \frac{1}{7}$

Step 4.3.4

Combine the numerators over the common denominator.

$K = \frac{3 \cdot 7 + 1}{7}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $3$ by $7$ .

$K = \frac{21 + 1}{7}$

Step 4.3.5.2

Add $21$ and $1$ .

$K = \frac{22}{7}$

Step 5

Substitute $\frac{22}{7}$ for $K$ in $r = - \frac{1}{14} \csc^{2} (7 θ) + K$ and simplify.

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Step 5.1

Substitute $\frac{22}{7}$ for $K$ .

$r = - \frac{1}{14} \csc^{2} (7 θ) + \frac{22}{7}$

Step 5.2

Combine $\csc^{2} (7 θ)$ and $\frac{1}{14}$ .

$r = - \frac{\csc^{2} (7 θ)}{14} + \frac{22}{7}$