Calculus Examples

$\ln (x) \frac{d x}{d y} = \frac{x}{y}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{x}$ .

$\frac{1}{x} (\ln (x) \frac{d x}{d y}) = \frac{1}{x} \cdot \frac{x}{y}$

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{x} (\ln (x) \frac{d x}{d y}) = \frac{1}{x} \cdot \frac{x}{y}$

Step 1.2.2

Rewrite the expression.

$\frac{1}{x} (\ln (x) \frac{d x}{d y}) = \frac{1}{y}$

Step 1.3

Remove unnecessary parentheses.

$\frac{1}{x} \ln (x) \frac{d x}{d y} = \frac{1}{y}$

Step 1.4

Rewrite the equation.

$\frac{1}{x} \ln (x) d x = \frac{1}{y} d y$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{x} \ln (x) d x = \int \frac{1}{y} d y$

Step 2.2

Integrate the left side.

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Step 2.2.1

Combine $\frac{1}{x}$ and $\ln (x)$ .

$\int \frac{\ln (x)}{x} d x = \int \frac{1}{y} d y$

Step 2.2.2

Let $u = \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.2.1

Let $u = \ln (x)$ . Find $\frac{d u}{d x}$ .

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Step 2.2.2.1.1

Differentiate $\ln (x)$ .

$\frac{d}{d x} [\ln (x)]$

Step 2.2.2.1.2

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

$\frac{1}{x}$

Step 2.2.2.2

Rewrite the problem using $u$ and $d u$ .

$\int u d u = \int \frac{1}{y} d y$

Step 2.2.3

By the Power Rule, the integral of $u$ with respect to $u$ is $\frac{1}{2} u^{2}$ .

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

Step 2.2.4

Replace all occurrences of $u$ with $\ln (x)$ .

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

Step 2.3

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

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

Step 2.4

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

$\frac{1}{2} \ln^{2} (x) = \ln (| y |) + C$