Calculus Examples

$(1 + x^{2}) d y - (1 + y^{2}) d x = 0$

Step 1

Add $(1 + y^{2}) d x$ to both sides of the equation.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $1 + x^{2}$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $1 + y^{2}$ .

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Step 3.2.1

Factor $1 + y^{2}$ out of $(1 + x^{2}) (1 + y^{2})$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

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 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

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

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

Step 5.2

Subtract $K$ from both sides of the equation.

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Add $K$ to both sides of the equation.

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

Step 5.7

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

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

Step 5.8

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 5.9

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

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

Step 5.10

Add $K$ to both sides of the equation.

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

Step 5.11

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

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