Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\sin (2 y)$ .

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

Step 1.2

Cancel the common factor of $\sin (2 y)$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

$\sin (2 y) d y = \cos (3 x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \sin (2 y) d y = \int \cos (3 x) d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u_{1} = 2 y$ . Then $d u_{1} = 2 d y$ , so $\frac{1}{2} d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.2.1.1

Let $u_{1} = 2 y$ . Find $\frac{d u_{1}}{d y}$ .

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Step 2.2.1.1.1

Differentiate $2 y$ .

$\frac{d}{d y} [2 y]$

Step 2.2.1.1.2

Since $2$ is constant with respect to $y$ , the derivative of $2 y$ with respect to $y$ is $2 \frac{d}{d y} [y]$ .

$2 \frac{d}{d y} [y]$

Step 2.2.1.1.3

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

$2 \cdot 1$

Step 2.2.1.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2.1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \sin (u_{1}) \frac{1}{2} d u_{1} = \int \cos (3 x) d x$

Step 2.2.2

Combine $\sin (u_{1})$ and $\frac{1}{2}$ .

$\int \frac{\sin (u_{1})}{2} d u_{1} = \int \cos (3 x) d x$

Step 2.2.3

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

$\frac{1}{2} \int \sin (u_{1}) d u_{1} = \int \cos (3 x) d x$

Step 2.2.4

The integral of $\sin (u_{1})$ with respect to $u_{1}$ is $- \cos (u_{1})$ .

$\frac{1}{2} (- \cos (u_{1}) + C_{1}) = \int \cos (3 x) d x$

Step 2.2.5

Simplify.

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Step 2.2.5.1

Simplify.

$\frac{1}{2} (- \cos (u_{1})) + C_{1} = \int \cos (3 x) d x$

Step 2.2.5.2

Combine $\frac{1}{2}$ and $\cos (u_{1})$ .

$- \frac{\cos (u_{1})}{2} + C_{1} = \int \cos (3 x) d x$

Step 2.2.6

Replace all occurrences of $u_{1}$ with $2 y$ .

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

Step 2.2.7

Reorder terms.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u_{2} = 3 x$ . Then $d u_{2} = 3 d x$ , so $\frac{1}{3} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.1.1

Let $u_{2} = 3 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.3.1.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 2.3.1.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 2.3.1.1.3

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

$3 \cdot 1$

Step 2.3.1.1.4

Multiply $3$ by $1$ .

$3$

Step 2.3.1.2

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

$- \frac{1}{2} \cos (2 y) + C_{1} = \int \cos (u_{2}) \frac{1}{3} d u_{2}$

Step 2.3.2

Combine $\cos (u_{2})$ and $\frac{1}{3}$ .

$- \frac{1}{2} \cos (2 y) + C_{1} = \int \frac{\cos (u_{2})}{3} d u_{2}$

Step 2.3.3

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

$- \frac{1}{2} \cos (2 y) + C_{1} = \frac{1}{3} \int \cos (u_{2}) d u_{2}$

Step 2.3.4

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$- \frac{1}{2} \cos (2 y) + C_{1} = \frac{1}{3} (\sin (u_{2}) + C_{2})$

Step 2.3.5

Simplify.

$- \frac{1}{2} \cos (2 y) + C_{1} = \frac{1}{3} \sin (u_{2}) + C_{2}$

Step 2.3.6

Replace all occurrences of $u_{2}$ with $3 x$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Apply the distributive property.

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

Step 3.2.1.2

Multiply $- 1$ by $1$ .

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

Step 3.2.1.3

Cancel the common factor of $2$ .

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Step 3.2.1.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 3.2.1.3.2

Factor $2$ out of $- 2 \sin^{2} (y)$ .

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

Step 3.2.1.3.3

Cancel the common factor.

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

Step 3.2.1.3.4

Rewrite the expression.

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

Step 3.2.1.4

Multiply.

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Step 3.2.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.1.4.2

Multiply $\sin^{2} (y)$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Apply the sine triple-angle identity.

$- \frac{1}{2} + \sin^{2} (y) = \frac{1}{3} (- 4 \sin^{3} (x) + 3 \sin (x)) + K$

Step 3.3.1.2

Apply the distributive property.

$- \frac{1}{2} + \sin^{2} (y) = \frac{1}{3} (- 4 \sin^{3} (x)) + \frac{1}{3} (3 \sin (x)) + K$

Step 3.3.1.3

Multiply $\frac{1}{3} (- 4 \sin^{3} (x))$ .

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Step 3.3.1.3.1

Combine $- 4$ and $\frac{1}{3}$ .

$- \frac{1}{2} + \sin^{2} (y) = \frac{- 4}{3} \sin^{3} (x) + \frac{1}{3} (3 \sin (x)) + K$

Step 3.3.1.3.2

Combine $\frac{- 4}{3}$ and $\sin^{3} (x)$ .

$- \frac{1}{2} + \sin^{2} (y) = \frac{- 4 \sin^{3} (x)}{3} + \frac{1}{3} (3 \sin (x)) + K$

Step 3.3.1.4

Cancel the common factor of $3$ .

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Step 3.3.1.4.1

Factor $3$ out of $3 \sin (x)$ .

$- \frac{1}{2} + \sin^{2} (y) = \frac{- 4 \sin^{3} (x)}{3} + \frac{1}{3} (3 (\sin (x))) + K$

Step 3.3.1.4.2

Cancel the common factor.

$- \frac{1}{2} + \sin^{2} (y) = \frac{- 4 \sin^{3} (x)}{3} + \frac{1}{3} (3 \sin (x)) + K$

Step 3.3.1.4.3

Rewrite the expression.

$- \frac{1}{2} + \sin^{2} (y) = \frac{- 4 \sin^{3} (x)}{3} + \sin (x) + K$

Step 3.3.1.5

Move the negative in front of the fraction.

$- \frac{1}{2} + \sin^{2} (y) = - \frac{4 \sin^{3} (x)}{3} + \sin (x) + K$

Step 3.4

Solve the equation for $y$ .

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Step 3.4.1

Add $\frac{1}{2}$ to both sides of the equation.

$\sin^{2} (y) = - \frac{4 \sin^{3} (x)}{3} + \sin (x) + K + \frac{1}{2}$

Step 3.4.2

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

$\sin (y) = \pm \sqrt{- \frac{4 \sin^{3} (x)}{3} + \sin (x) + K + \frac{1}{2}}$

Step 3.4.3

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

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Step 3.4.3.1

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

$\sin (y) = \sqrt{- \frac{4 \sin^{3} (x)}{3} + \sin (x) + K + \frac{1}{2}}$

Step 3.4.3.2

Take the inverse sine of both sides of the equation to extract $y$ from inside the sine.

$y = \arcsin (\sqrt{- \frac{4 \sin^{3} (x)}{3} + \sin (x) + K + \frac{1}{2}})$

Step 3.4.3.3

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

$\sin (y) = - \sqrt{- \frac{4 \sin^{3} (x)}{3} + \sin (x) + K + \frac{1}{2}}$

Step 3.4.3.4

Take the inverse sine of both sides of the equation to extract $y$ from inside the sine.

$y = \arcsin (- \sqrt{- \frac{4 \sin^{3} (x)}{3} + \sin (x) + K + \frac{1}{2}})$

Step 3.4.3.5