Calculus Examples

$\int \ln (\sqrt[3]{x}) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (\sqrt[3]{x})$ and $d v = 1$ .

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

Step 2

Simplify.

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Step 2.1

Combine $x$ and $\frac{1}{3 x}$ .

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

Apply the constant rule.

$\ln (\sqrt[3]{x}) x - (\frac{1}{3} x + C)$

Step 4

Rewrite $\ln (\sqrt[3]{x}) x - (\frac{1}{3} x + C)$ as $\ln (\sqrt[3]{x}) x - \frac{1}{3} x + C$ .

$\ln (\sqrt[3]{x}) x - \frac{1}{3} x + C$