Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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.

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

Step 2.2.1.2

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

$\int 1 \sin (y) d y = \int \cos (x) d x$

Step 2.2.1.3

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

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 2.4

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

$- \cos (y) = \sin (x) + K$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sin (x) + K = - \cos (y)$ .

$\sin (x) + K = - \cos (y)$

Step 3.2

Subtract $K$ from both sides of the equation.

$\sin (x) = - \cos (y) - K$

Step 3.3

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

$x = \arcsin (- \cos (y) - K)$

Step 3.4

Rewrite the equation as $\arcsin (- \cos (y) - K) = x$ .

$\arcsin (- \cos (y) - K) = x$

Step 3.5

Take the inverse arcsine of both sides of the equation to extract $\cos (y)$ from inside the arcsine.

$- \cos (y) - K = \sin (x)$

Step 3.6

Add $K$ to both sides of the equation.

$- \cos (y) = \sin (x) + K$

Step 3.7

Divide each term in $- \cos (y) = \sin (x) + K$ by $- 1$ and simplify.

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Step 3.7.1

Divide each term in $- \cos (y) = \sin (x) + K$ by $- 1$ .

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

Step 3.7.2

Simplify the left side.

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Step 3.7.2.1

Dividing two negative values results in a positive value.

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

Step 3.7.2.2

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

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

Step 3.7.3

Simplify the right side.

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Step 3.7.3.1

Simplify each term.

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Step 3.7.3.1.1

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

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

Step 3.7.3.1.2

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

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

Step 3.7.3.1.3

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

$\cos (y) = - \sin (x) - 1 \cdot K$

Step 3.7.3.1.4

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

$\cos (y) = - \sin (x) - K$

Step 3.8

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

$y = \arccos (- \sin (x) - K)$

Step 3.9

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\arcsin (- \cos (y) - K) = x$

Step 3.10

Take the inverse arcsine of both sides of the equation to extract $\cos (y)$ from inside the arcsine.

$- \cos (y) - K = \sin (x)$

Step 3.11

Add $K$ to both sides of the equation.

$- \cos (y) = \sin (x) + K$

Step 3.12

Divide each term in $- \cos (y) = \sin (x) + K$ by $- 1$ and simplify.

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Step 3.12.1

Divide each term in $- \cos (y) = \sin (x) + K$ by $- 1$ .

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

Step 3.12.2

Simplify the left side.

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Step 3.12.2.1

Dividing two negative values results in a positive value.

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

Step 3.12.2.2

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

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

Step 3.12.3

Simplify the right side.

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Step 3.12.3.1

Simplify each term.

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Step 3.12.3.1.1

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

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

Step 3.12.3.1.2

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

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

Step 3.12.3.1.3

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

$\cos (y) = - \sin (x) - 1 \cdot K$

Step 3.12.3.1.4

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

$\cos (y) = - \sin (x) - K$

Step 3.13

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

$y = \arccos (- \sin (x) - K)$

Step 4

Simplify the constant of integration.

$y = \arccos (- \sin (x) + K)$