Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the fraction into multiple fractions.

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

Cancel the common factor of $x$ .

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Step 2.3.3.1

Cancel the common factor.

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

Step 2.3.3.2

Rewrite the expression.

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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

Step 2.4

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

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