Calculus Examples

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

Step 1

Rewrite the equation.

$d r = π \cos (π θ) d θ$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d r = \int π \cos (π θ) d θ$

Step 2.2

Apply the constant rule.

$r + C_{1} = \int π \cos (π θ) d θ$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$r + C_{1} = π \int \cos (π θ) d θ$

Step 2.3.2

Let $u = π θ$ . Then $d u = π d θ$ , so $\frac{1}{π} d u = d θ$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = π θ$ . Find $\frac{d u}{d θ}$ .

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Step 2.3.2.1.1

Differentiate $π θ$ .

$\frac{d}{d θ} [π θ]$

Step 2.3.2.1.2

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

$π \frac{d}{d θ} [θ]$

Step 2.3.2.1.3

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

$π \cdot 1$

Step 2.3.2.1.4

Multiply $π$ by $1$ .

$π$

Step 2.3.2.2

Rewrite the problem using $u$ and $d u$ .

$r + C_{1} = π \int \cos (u) \frac{1}{π} d u$

Step 2.3.3

Combine $\cos (u)$ and $\frac{1}{π}$ .

$r + C_{1} = π \int \frac{\cos (u)}{π} d u$

Step 2.3.4

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

$r + C_{1} = π (\frac{1}{π} \int \cos (u) d u)$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Combine $\frac{1}{π}$ and $π$ .

$r + C_{1} = \frac{π}{π} \int \cos (u) d u$

Step 2.3.5.2

Cancel the common factor of $π$ .

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Step 2.3.5.2.1

Cancel the common factor.

$r + C_{1} = \frac{π}{π} \int \cos (u) d u$

Step 2.3.5.2.2

Rewrite the expression.

$r + C_{1} = 1 \int \cos (u) d u$

Step 2.3.5.3

Multiply $\int \cos (u) d u$ by $1$ .

$r + C_{1} = \int \cos (u) d u$

Step 2.3.6

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$r + C_{1} = \sin (u) + C_{2}$

Step 2.3.7

Replace all occurrences of $u$ with $π θ$ .

$r + C_{1} = \sin (π θ) + C_{2}$

Step 2.4

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

$r = \sin (π θ) + K$

Step 3

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

$2 = \sin (π \cdot 2) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\sin (π \cdot 2) + K = 2$ .

$\sin (π \cdot 2) + K = 2$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $\sin (π \cdot 2) + K$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Move $2$ to the left of $π$ .

$\sin (2 \cdot π) + K = 2$

Step 4.2.1.1.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sin (0) + K = 2$

Step 4.2.1.1.3

The exact value of $\sin (0)$ is $0$ .

$0 + K = 2$

Step 4.2.1.2

Add $0$ and $K$ .

$K = 2$

Step 5

Substitute $2$ for $K$ in $r = \sin (π θ) + K$ and simplify.

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Step 5.1

Substitute $2$ for $K$ .

$r = \sin (π θ) + 2$