Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the fraction into multiple fractions.

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

Cancel the common factor of $x$ .

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Step 2.3.3.1.1

Cancel the common factor.

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

Step 2.3.3.1.2

Rewrite the expression.

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

Step 2.3.3.2

Move the negative in front of the fraction.

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.3.6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.7

Multiply $2$ by $- 1$ .

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

Step 2.3.8

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

$y + C_{1} = x + C_{2} - 2 (\ln (| x |) + C_{3})$

Step 2.3.9

Simplify.

$y + C_{1} = x - 2 \ln (| x |) + C_{4}$

Step 2.4

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

$y = x - 2 \ln (| x |) + C$