Calculus Examples

$4 \ln (x)$

Step 1

Write $4 \ln (x)$ as a function.

$f (x) = 4 \ln (x)$

Step 2

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 3

Set up the integral to solve.

$F (x) = \int 4 \ln (x) d x$

Step 4

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int \ln (x) d x$

Step 5

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

$4 (\ln (x) x - \int x \frac{1}{x} d x)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

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

$4 (\ln (x) x - \int \frac{x}{x} d x)$

Step 6.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 6.2.1

Cancel the common factor.

$4 (\ln (x) x - \int \frac{x}{x} d x)$

Step 6.2.2

Rewrite the expression.

$4 (\ln (x) x - \int d x)$

Step 7

Apply the constant rule.

$4 (\ln (x) x - (x + C))$

Step 8

Simplify.

$4 (\ln (x) x - x) + C$

Step 9

The answer is the antiderivative of the function $f (x) = 4 \ln (x)$ .

$F (x) =$ $4 (\ln (x) x - x) + C$