Calculus Examples

$2 x \frac{d y}{d x} - \ln (x^{2}) = 0$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Add $\ln (x^{2})$ to both sides of the equation.

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

Step 1.1.2

Divide each term in $2 x \frac{d y}{d x} = \ln (x^{2})$ by $2 x$ and simplify.

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Step 1.1.2.1

Divide each term in $2 x \frac{d y}{d x} = \ln (x^{2})$ by $2 x$ .

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $2$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

Rewrite the expression.

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

Step 1.1.2.2.2

Cancel the common factor of $x$ .

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Step 1.1.2.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Expand $\ln (x^{2})$ by moving $2$ outside the logarithm.

$\frac{d y}{d x} = \frac{2 \ln (x)}{2 x}$

Step 1.1.2.3.2

Cancel the common factor of $2$ .

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Step 1.1.2.3.2.1

Cancel the common factor.

$\frac{d y}{d x} = \frac{2 \ln (x)}{2 x}$

Step 1.1.2.3.2.2

Rewrite the expression.

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

Step 1.2

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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 2.3.1.1

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

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Step 2.3.1.1.1

Differentiate $\ln (x)$ .

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

Step 2.3.1.1.2

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

$\frac{1}{x}$

Step 2.3.1.2

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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