Calculus Examples

$\frac{d y}{d x} = \frac{3 \sin (π x)}{2 y}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y \frac{3 \sin (π x)}{2 y}$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Factor $y$ out of $2 y$ .

$y \frac{d y}{d x} = y \frac{3 \sin (π x)}{y \cdot 2}$

Step 1.2.2

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{3 \sin (π x)}{y \cdot 2}$

Step 1.2.3

Rewrite the expression.

$y \frac{d y}{d x} = \frac{3 \sin (π x)}{2}$

Step 1.3

Rewrite the equation.

$y d y = \frac{3 \sin (π x)}{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{3 \sin (π x)}{2} d x$

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{3 \sin (π x)}{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{3}{2}$ is constant with respect to $x$ , move $\frac{3}{2}$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = \frac{3}{2} \int \sin (π x) d x$

Step 2.3.2

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

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Differentiate $π x$ .

$\frac{d}{d x} [π x]$

Step 2.3.2.1.2

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

$π \frac{d}{d x} [x]$

Step 2.3.2.1.3

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

$\frac{1}{2} y^{2} + C_{1} = \frac{3}{2} \int \sin (u) \frac{1}{π} d u$

Step 2.3.3

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

$\frac{1}{2} y^{2} + C_{1} = \frac{3}{2} \int \frac{\sin (u)}{π} d u$

Step 2.3.4

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

$\frac{1}{2} y^{2} + C_{1} = \frac{3}{2} (\frac{1}{π} \int \sin (u) d u)$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Multiply $\frac{1}{π}$ by $\frac{3}{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{3}{π \cdot 2} \int \sin (u) d u$

Step 2.3.5.2

Move $2$ to the left of $π$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{3}{2 π} \int \sin (u) d u$

Step 2.3.6

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

$\frac{1}{2} y^{2} + C_{1} = \frac{3}{2 π} (- \cos (u) + C_{2})$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

$\frac{1}{2} y^{2} + C_{1} = \frac{3}{2 π} (- \cos (u)) + C_{2}$

Step 2.3.7.2

Combine $\frac{3}{2 π}$ and $\cos (u)$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{3 \cos (u)}{2 π} + C_{2}$

Step 2.3.8

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

$\frac{1}{2} y^{2} + C_{1} = - \frac{3 \cos (π x)}{2 π} + C_{2}$

Step 2.3.9

Reorder terms.

$\frac{1}{2} y^{2} + C_{1} = - \frac{3}{2 π} \cos (π x) + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = - \frac{3}{2 π} \cos (π x) + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (- \frac{3}{2 π} \cos (π x) + K)$

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 $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

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

$2 \frac{y^{2}}{2} = 2 (- \frac{3}{2 π} \cos (π x) + K)$

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.

$2 \frac{y^{2}}{2} = 2 (- \frac{3}{2 π} \cos (π x) + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (- \frac{3}{2 π} \cos (π x) + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (- \frac{3}{2 π} \cos (π x) + K)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Combine $\cos (π x)$ and $\frac{3}{2 π}$ .

$y^{2} = 2 (- \frac{\cos (π x) \cdot 3}{2 π} + K)$

Step 3.2.2.1.1.2

Move $3$ to the left of $\cos (π x)$ .

$y^{2} = 2 (- \frac{3 \cos (π x)}{2 π} + K)$

Step 3.2.2.1.2

Simplify terms.

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Step 3.2.2.1.2.1

Apply the distributive property.

$y^{2} = 2 (- \frac{3 \cos (π x)}{2 π}) + 2 K$

Step 3.2.2.1.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.2.2.1

Move the leading negative in $- \frac{3 \cos (π x)}{2 π}$ into the numerator.

$y^{2} = 2 \frac{- 3 \cos (π x)}{2 π} + 2 K$

Step 3.2.2.1.2.2.2

Factor $2$ out of $2 π$ .

$y^{2} = 2 \frac{- 3 \cos (π x)}{2 (π)} + 2 K$

Step 3.2.2.1.2.2.3

Cancel the common factor.

$y^{2} = 2 \frac{- 3 \cos (π x)}{2 π} + 2 K$

Step 3.2.2.1.2.2.4

Rewrite the expression.

$y^{2} = \frac{- 3 \cos (π x)}{π} + 2 K$

Step 3.2.2.1.2.3

Move the negative in front of the fraction.

$y^{2} = - \frac{3 \cos (π x)}{π} + 2 K$

Step 3.3

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

$y = \pm \sqrt{- \frac{3 \cos (π x)}{π} + 2 K}$

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{- \frac{3 \cos (π x)}{π} + 2 K}$

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{- \frac{3 \cos (π x)}{π} + 2 K}$

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{- \frac{3 \cos (π x)}{π} + 2 K}$

$y = - \sqrt{- \frac{3 \cos (π x)}{π} + 2 K}$

$y = \sqrt{- \frac{3 \cos (π x)}{π} + 2 K}$

$y = - \sqrt{- \frac{3 \cos (π x)}{π} + 2 K}$

$y = \sqrt{- \frac{3 \cos (π x)}{π} + 2 K}$

$y = - \sqrt{- \frac{3 \cos (π x)}{π} + 2 K}$

Step 4

Simplify the constant of integration.

$y = \sqrt{- \frac{3 \cos (π x)}{π} + K}$

$y = - \sqrt{- \frac{3 \cos (π x)}{π} + K}$