Calculus Examples

$\ln (x^{3})$

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

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

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

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

Set up the integral to solve.

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

Rewrite as $\int 3 \ln (x) d x$ .

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

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

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

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

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

Simplify.

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

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

Cancel the common factor of $x$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Apply the constant rule.

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

Simplify.

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

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

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