Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\cos^{2} (y)}$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Combine.

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

Step 1.2.2

Cancel the common factor of $\cos^{2} (y)$ .

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

Rewrite the expression.

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

Step 1.2.3

Rewrite $1$ as $1^{2}$ .

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Convert from $\frac{1}{\cos^{2} (y)}$ to $\sec^{2} (y)$ .

$\int \sec^{2} (y) d y = \int \csc^{2} (x) d x$

Step 2.2.2

Since the derivative of $\tan (y)$ is $\sec^{2} (y)$ , the integral of $\sec^{2} (y)$ is $\tan (y)$ .

$\tan (y) + C_{1} = \int \csc^{2} (x) d x$

Step 2.3

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

$\tan (y) + C_{1} = - \cot (x) + C_{2}$

Step 2.4

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

$\tan (y) = - \cot (x) + K$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- \cot (x) + K = \tan (y)$ .

$- \cot (x) + K = \tan (y)$

Step 3.2

Subtract $K$ from both sides of the equation.

$- \cot (x) = \tan (y) - K$

Step 3.3

Divide each term in $- \cot (x) = \tan (y) - K$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- \cot (x) = \tan (y) - K$ by $- 1$ .

$\frac{- \cot (x)}{- 1} = \frac{\tan (y)}{- 1} + \frac{- K}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{\cot (x)}{1} = \frac{\tan (y)}{- 1} + \frac{- K}{- 1}$

Step 3.3.2.2

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

$\cot (x) = \frac{\tan (y)}{- 1} + \frac{- K}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

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

$\cot (x) = - 1 \cdot \tan (y) + \frac{- K}{- 1}$

Step 3.3.3.1.2

Rewrite $- 1 \cdot \tan (y)$ as $- \tan (y)$ .

$\cot (x) = - \tan (y) + \frac{- K}{- 1}$

Step 3.3.3.1.3

Dividing two negative values results in a positive value.

$\cot (x) = - \tan (y) + \frac{K}{1}$

Step 3.3.3.1.4

Divide $K$ by $1$ .

$\cot (x) = - \tan (y) + K$

Step 3.4

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

$x = arccot (- \tan (y) + K)$

Step 3.5

Rewrite the equation as $arccot (- \tan (y) + K) = x$ .

$arccot (- \tan (y) + K) = x$

Step 3.6

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

$- \tan (y) + K = \cot (x)$

Step 3.7

Subtract $K$ from both sides of the equation.

$- \tan (y) = \cot (x) - K$

Step 3.8

Divide each term in $- \tan (y) = \cot (x) - K$ by $- 1$ and simplify.

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Step 3.8.1

Divide each term in $- \tan (y) = \cot (x) - K$ by $- 1$ .

$\frac{- \tan (y)}{- 1} = \frac{\cot (x)}{- 1} + \frac{- K}{- 1}$

Step 3.8.2

Simplify the left side.

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Step 3.8.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (y)}{1} = \frac{\cot (x)}{- 1} + \frac{- K}{- 1}$

Step 3.8.2.2

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

$\tan (y) = \frac{\cot (x)}{- 1} + \frac{- K}{- 1}$

Step 3.8.3

Simplify the right side.

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Step 3.8.3.1

Simplify each term.

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Step 3.8.3.1.1

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

$\tan (y) = - 1 \cdot \cot (x) + \frac{- K}{- 1}$

Step 3.8.3.1.2

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

$\tan (y) = - \cot (x) + \frac{- K}{- 1}$

Step 3.8.3.1.3

Dividing two negative values results in a positive value.

$\tan (y) = - \cot (x) + \frac{K}{1}$

Step 3.8.3.1.4

Divide $K$ by $1$ .

$\tan (y) = - \cot (x) + K$

Step 3.9

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

$y = \arctan (- \cot (x) + K)$

Step 3.10

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

$arccot (- \tan (y) + K) = x$

Step 3.11

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

$- \tan (y) + K = \cot (x)$

Step 3.12

Subtract $K$ from both sides of the equation.

$- \tan (y) = \cot (x) - K$

Step 3.13

Divide each term in $- \tan (y) = \cot (x) - K$ by $- 1$ and simplify.

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Step 3.13.1

Divide each term in $- \tan (y) = \cot (x) - K$ by $- 1$ .

$\frac{- \tan (y)}{- 1} = \frac{\cot (x)}{- 1} + \frac{- K}{- 1}$

Step 3.13.2

Simplify the left side.

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Step 3.13.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (y)}{1} = \frac{\cot (x)}{- 1} + \frac{- K}{- 1}$

Step 3.13.2.2

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

$\tan (y) = \frac{\cot (x)}{- 1} + \frac{- K}{- 1}$

Step 3.13.3

Simplify the right side.

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Step 3.13.3.1

Simplify each term.

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Step 3.13.3.1.1

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

$\tan (y) = - 1 \cdot \cot (x) + \frac{- K}{- 1}$

Step 3.13.3.1.2

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

$\tan (y) = - \cot (x) + \frac{- K}{- 1}$

Step 3.13.3.1.3

Dividing two negative values results in a positive value.

$\tan (y) = - \cot (x) + \frac{K}{1}$

Step 3.13.3.1.4

Divide $K$ by $1$ .

$\tan (y) = - \cot (x) + K$

Step 3.14

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

$y = \arctan (- \cot (x) + K)$