Calculus Examples

$\frac{d y}{d x} = \frac{x}{\sin (y)} - \frac{9}{\sin (y)}$

Step 1

Separate the variables.

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Step 1.1

Factor.

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Step 1.1.1

Separate fractions.

$\frac{d y}{d x} = \frac{x}{\sin (y)} - (\frac{9}{1} \cdot \frac{1}{\sin (y)})$

Step 1.1.2

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

$\frac{d y}{d x} = \frac{x}{\sin (y)} - (\frac{9}{1} \csc (y))$

Step 1.1.3

Divide $9$ by $1$ .

$\frac{d y}{d x} = \frac{x}{\sin (y)} - (9 \csc (y))$

Step 1.1.4

Multiply $9$ by $- 1$ .

$\frac{d y}{d x} = \frac{x}{\sin (y)} - 9 \csc (y)$

Step 1.1.5

Simplify each term.

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Step 1.1.5.1

Separate fractions.

$\frac{d y}{d x} = \frac{x}{1} \cdot \frac{1}{\sin (y)} - 9 \csc (y)$

Step 1.1.5.2

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

$\frac{d y}{d x} = \frac{x}{1} \csc (y) - 9 \csc (y)$

Step 1.1.5.3

Divide $x$ by $1$ .

$\frac{d y}{d x} = x \csc (y) - 9 \csc (y)$

Step 1.1.6

Factor $\csc (y)$ out of $x \csc (y) - 9 \csc (y)$ .

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Step 1.1.6.1

Factor $\csc (y)$ out of $x \csc (y)$ .

$\frac{d y}{d x} = \csc (y) x - 9 \csc (y)$

Step 1.1.6.2

Factor $\csc (y)$ out of $- 9 \csc (y)$ .

$\frac{d y}{d x} = \csc (y) x + \csc (y) \cdot - 9$

Step 1.1.6.3

Factor $\csc (y)$ out of $\csc (y) x + \csc (y) \cdot - 9$ .

$\frac{d y}{d x} = \csc (y) (x - 9)$

Step 1.2

Multiply both sides by $\frac{1}{\csc (y)}$ .

$\frac{1}{\csc (y)} \frac{d y}{d x} = \frac{1}{\csc (y)} (\csc (y) (x - 9))$

Step 1.3

Cancel the common factor of $\csc (y)$ .

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Step 1.3.1

Cancel the common factor.

$\frac{1}{\csc (y)} \frac{d y}{d x} = \frac{1}{\csc (y)} (\csc (y) (x - 9))$

Step 1.3.2

Rewrite the expression.

$\frac{1}{\csc (y)} \frac{d y}{d x} = x - 9$

Step 1.4

Rewrite the equation.

$\frac{1}{\csc (y)} d y = (x - 9) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\csc (y)} d y = \int x - 9 d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

Rewrite $\csc (y)$ in terms of sines and cosines.

$\int \frac{1}{\frac{1}{\sin (y)}} d y = \int x - 9 d x$

Step 2.2.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (y)}$ .

$\int 1 \sin (y) d y = \int x - 9 d x$

Step 2.2.1.3

Multiply $\sin (y)$ by $1$ .

$\int \sin (y) d y = \int x - 9 d x$

Step 2.2.2

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

$- \cos (y) + C_{1} = \int x - 9 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$- \cos (y) + C_{1} = \int x d x + \int - 9 d x$

Step 2.3.2

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

$- \cos (y) + C_{1} = \frac{1}{2} x^{2} + C_{2} + \int - 9 d x$

Step 2.3.3

Apply the constant rule.

$- \cos (y) + C_{1} = \frac{1}{2} x^{2} + C_{2} - 9 x + C_{3}$

Step 2.3.4

Simplify.

$- \cos (y) + C_{1} = \frac{1}{2} x^{2} - 9 x + C_{4}$

Step 2.4

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

$- \cos (y) = \frac{1}{2} x^{2} - 9 x + K$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $- \cos (y) = \frac{1}{2} x^{2} - 9 x + K$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- \cos (y) = \frac{1}{2} x^{2} - 9 x + K$ by $- 1$ .

$\frac{- \cos (y)}{- 1} = \frac{\frac{1}{2} x^{2}}{- 1} + \frac{- 9 x}{- 1} + \frac{K}{- 1}$

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.

$\frac{\cos (y)}{1} = \frac{\frac{1}{2} x^{2}}{- 1} + \frac{- 9 x}{- 1} + \frac{K}{- 1}$

Step 3.1.2.2

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

$\cos (y) = \frac{\frac{1}{2} x^{2}}{- 1} + \frac{- 9 x}{- 1} + \frac{K}{- 1}$

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 $\frac{\frac{1}{2} x^{2}}{- 1}$ .

$\cos (y) = - 1 \cdot (\frac{1}{2} x^{2}) + \frac{- 9 x}{- 1} + \frac{K}{- 1}$

Step 3.1.3.1.2

Rewrite $- 1 \cdot (\frac{1}{2} x^{2})$ as $- (\frac{1}{2} x^{2})$ .

$\cos (y) = - (\frac{1}{2} x^{2}) + \frac{- 9 x}{- 1} + \frac{K}{- 1}$

Step 3.1.3.1.3

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

$\cos (y) = - \frac{x^{2}}{2} + \frac{- 9 x}{- 1} + \frac{K}{- 1}$

Step 3.1.3.1.4

Move the negative one from the denominator of $\frac{- 9 x}{- 1}$ .

$\cos (y) = - \frac{x^{2}}{2} - 1 \cdot (- 9 x) + \frac{K}{- 1}$

Step 3.1.3.1.5

Rewrite $- 1 \cdot (- 9 x)$ as $- (- 9 x)$ .

$\cos (y) = - \frac{x^{2}}{2} - (- 9 x) + \frac{K}{- 1}$

Step 3.1.3.1.6

Multiply $- 9$ by $- 1$ .

$\cos (y) = - \frac{x^{2}}{2} + 9 x + \frac{K}{- 1}$

Step 3.1.3.1.7

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

$\cos (y) = - \frac{x^{2}}{2} + 9 x - 1 \cdot K$

Step 3.1.3.1.8

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

$\cos (y) = - \frac{x^{2}}{2} + 9 x - K$

Step 3.2

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

$y = \arccos (- \frac{x^{2}}{2} + 9 x - K)$

Step 4

Simplify the constant of integration.

$y = \arccos (- \frac{x^{2}}{2} + 9 x + K)$