Calculus Examples

$\int \ln (y) d y$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

$\ln (y) y - \int \frac{y}{y} d y$

Step 2.2

Cancel the common factor of $y$ .

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Step 2.2.1

Cancel the common factor.

$\ln (y) y - \int \frac{y}{y} d y$

Step 2.2.2

Rewrite the expression.

$\ln (y) y - \int d y$

Step 3

Apply the constant rule.

$\ln (y) y - (y + C)$

Step 4

Simplify.

$\ln (y) y - y + C$