Calculus Examples

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

Step 1

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

$d y = - \frac{1}{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

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.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify.

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

Step 2.4

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

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