Calculus Examples

$\frac{1}{x^{2} + 1} d x + \frac{1}{y^{2} + 1} d y = 0$

Step 1

Subtract $\frac{1}{x^{2} + 1} d x$ from both sides of the equation.

$\frac{1}{y^{2} + 1} d y = - \frac{1}{x^{2} + 1} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{2} + 1} d y = \int - \frac{1}{x^{2} + 1} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

Reorder $y^{2}$ and $1$ .

$\int \frac{1}{1 + y^{2}} d y = \int - \frac{1}{x^{2} + 1} d x$

Step 2.2.1.2

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

$\int \frac{1}{1^{2} + y^{2}} d y = \int - \frac{1}{x^{2} + 1} d x$

Step 2.2.2

The integral of $\frac{1}{1^{2} + y^{2}}$ with respect to $y$ is $\arctan (y) + C_{1}$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

Reorder $x^{2}$ and $1$ .

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

Step 2.3.2.2

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

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

Step 2.3.3

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C_{2}$ .

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

Step 2.3.4

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Subtract $K$ from both sides of the equation.

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

Step 3.3

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

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Step 3.3.1

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

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

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.

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

Step 3.3.2.2

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

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

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 $\frac{\arctan (y)}{- 1}$ .

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

Step 3.3.3.1.2

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

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

Step 3.3.3.1.3

Dividing two negative values results in a positive value.

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

Step 3.3.3.1.4

Divide $K$ by $1$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Subtract $K$ from both sides of the equation.

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

Step 3.8

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

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Step 3.8.1

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

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

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.

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

Step 3.8.2.2

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

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

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 $\frac{\arctan (x)}{- 1}$ .

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

Step 3.8.3.1.2

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

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

Step 3.8.3.1.3

Dividing two negative values results in a positive value.

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

Step 3.8.3.1.4

Divide $K$ by $1$ .

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

Step 3.9

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

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

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.

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

Step 3.11

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

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

Step 3.12

Subtract $K$ from both sides of the equation.

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

Step 3.13

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

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Step 3.13.1

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

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

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.

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

Step 3.13.2.2

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

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

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 $\frac{\arctan (x)}{- 1}$ .

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

Step 3.13.3.1.2

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

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

Step 3.13.3.1.3

Dividing two negative values results in a positive value.

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

Step 3.13.3.1.4

Divide $K$ by $1$ .

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

Step 3.14

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

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