Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

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

Step 4.4

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

$\arctan (y) = \ln (| x |) + C$

Step 5

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

$y = \tan (\ln (| x |) + C)$