Calculus Examples

$\ln ({(x)}^{3})$

Step 1

Write $\ln ({(x)}^{3})$ as a function.

$f (x) = \ln ({(x)}^{3})$

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.

$F (x) = \int \ln ({(x)}^{3}) d x$

Step 4

Rewrite $\ln ({(x)}^{3})$ as $3 \ln (x)$ .

$\int 3 \ln (x) d x$

Step 5

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int \ln (x) d x$

Step 6

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

$3 (\ln (x) x - \int x \frac{1}{x} d x)$

Step 7

Simplify.

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Step 7.1

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

$3 (\ln (x) x - \int \frac{x}{x} d x)$

Step 7.2

Cancel the common factor of $x$ .

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Step 7.2.1

Cancel the common factor.

$3 (\ln (x) x - \int \frac{x}{x} d x)$

Step 7.2.2

Rewrite the expression.

$3 (\ln (x) x - \int d x)$

Step 8

Apply the constant rule.

$3 (\ln (x) x - (x + C))$

Step 9

Simplify.

$3 (\ln (x) x - x) + C$

Step 10

The answer is the antiderivative of the function $f (x) = \ln ({(x)}^{3})$ .

$F (x) =$ $3 (\ln (x) x - x) + C$