Calculus Examples

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

Step 1

Rewrite the equation.

$d y = \ln (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \ln (x) d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = 1$ .

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Combine $x$ and $\frac{1}{x}$ .

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

Step 2.3.2.2

Cancel the common factor of $x$ .

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Step 2.3.2.2.1

Cancel the common factor.

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

Step 2.3.2.2.2

Rewrite the expression.

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

Step 2.3.3

Apply the constant rule.

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

Step 2.3.4

Simplify.

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

Step 2.3.5

Reorder terms.

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

Step 2.4

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

$y = x \ln (x) - x + K$