Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

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

Step 1.2

Multiply both sides by $\sin (x)$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\sin (x) \frac{d x}{d y} = \sin (x) (\frac{1 + 2 y^{2}}{y} \csc (x))$

Step 1.3.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.3.2.1

Reorder $\sin (x)$ and $\frac{1 + 2 y^{2}}{y} \csc (x)$ .

$\sin (x) \frac{d x}{d y} = \frac{1 + 2 y^{2}}{y} \csc (x) \sin (x)$

Step 1.3.2.2

Add parentheses.

$\sin (x) \frac{d x}{d y} = \frac{1 + 2 y^{2}}{y} (\csc (x) \sin (x))$

Step 1.3.2.3

Rewrite $\sin (x) (\frac{1 + 2 y^{2}}{y} \csc (x))$ in terms of sines and cosines.

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

Step 1.3.2.4

Cancel the common factors.

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

Step 1.3.3

Multiply $\frac{1 + 2 y^{2}}{y}$ by $1$ .

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the fraction into multiple fractions.

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

Cancel the common factor of $y^{2}$ and $y$ .

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Step 2.3.3.1

Factor $y$ out of $2 y^{2}$ .

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

Step 2.3.3.2

Cancel the common factors.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

Factor $y$ out of $y^{1}$ .

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

Step 2.3.3.2.3

Cancel the common factor.

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

Step 2.3.3.2.4

Rewrite the expression.

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

Step 2.3.3.2.5

Divide $2 y$ by $1$ .

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

Step 2.3.4

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

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

Step 2.3.5

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

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

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

$- \cos (x) + C_{1} = \ln (| y |) + 2 (\frac{1}{2}) y^{2} + C_{4}$

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

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

$- \cos (x) + C_{1} = \ln (| y |) + \frac{2}{2} y^{2} + C_{4}$

Step 2.3.7.2.2

Cancel the common factor of $2$ .

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Step 2.3.7.2.2.1

Cancel the common factor.

$- \cos (x) + C_{1} = \ln (| y |) + \frac{2}{2} y^{2} + C_{4}$

Step 2.3.7.2.2.2

Rewrite the expression.

$- \cos (x) + C_{1} = \ln (| y |) + 1 y^{2} + C_{4}$

Step 2.3.7.2.3

Multiply $y^{2}$ by $1$ .

$- \cos (x) + C_{1} = \ln (| y |) + y^{2} + C_{4}$

Step 2.3.8

Reorder terms.

$- \cos (x) + C_{1} = y^{2} + \ln (| y |) + C_{4}$

Step 2.4

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

$- \cos (x) = y^{2} + \ln (| y |) + C$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $- \cos (x) = y^{2} + \ln (| y |) + C$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- \cos (x) = y^{2} + \ln (| y |) + C$ by $- 1$ .

$\frac{- \cos (x)}{- 1} = \frac{y^{2}}{- 1} + \frac{\ln (| y |)}{- 1} + \frac{C}{- 1}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Dividing two negative values results in a positive value.

$\frac{\cos (x)}{1} = \frac{y^{2}}{- 1} + \frac{\ln (| y |)}{- 1} + \frac{C}{- 1}$

Step 3.1.2.2

Divide $\cos (x)$ by $1$ .

$\cos (x) = \frac{y^{2}}{- 1} + \frac{\ln (| y |)}{- 1} + \frac{C}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Move the negative one from the denominator of $\frac{y^{2}}{- 1}$ .

$\cos (x) = - 1 \cdot y^{2} + \frac{\ln (| y |)}{- 1} + \frac{C}{- 1}$

Step 3.1.3.1.2

Rewrite $- 1 \cdot y^{2}$ as $- y^{2}$ .

$\cos (x) = - y^{2} + \frac{\ln (| y |)}{- 1} + \frac{C}{- 1}$

Step 3.1.3.1.3

Move the negative one from the denominator of $\frac{\ln (| y |)}{- 1}$ .

$\cos (x) = - y^{2} - 1 \cdot \ln (| y |) + \frac{C}{- 1}$

Step 3.1.3.1.4

Rewrite $- 1 \cdot \ln (| y |)$ as $- \ln (| y |)$ .

$\cos (x) = - y^{2} - \ln (| y |) + \frac{C}{- 1}$

Step 3.1.3.1.5

Move the negative one from the denominator of $\frac{C}{- 1}$ .

$\cos (x) = - y^{2} - \ln (| y |) - 1 \cdot C$

Step 3.1.3.1.6

Rewrite $- 1 \cdot C$ as $- C$ .

$\cos (x) = - y^{2} - \ln (| y |) - C$

Step 3.2

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (- y^{2} - \ln (| y |) - C)$

Step 4

Simplify the constant of integration.

$x = \arccos (- y^{2} - \ln (| y |) + C)$