Calculus Examples

$f (x) = \ln (x)$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Cancel the common factor of $x$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 5

Apply the constant rule.

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

Step 6

Simplify.

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

Step 7

The answer is the antiderivative of the function $f (x) = \ln (x)$ .

$F (x) =$ $\ln (x) x - x + C$