Calculus Examples

$\int 16 \ln (\sqrt[3]{x}) d x$

Step 1

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$16 \int \ln (\sqrt[3]{x}) d x$

Step 2

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

$16 (\ln (\sqrt[3]{x}) x - \int x \frac{1}{3 x} d x)$

Step 3

Simplify.

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Step 3.1

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

$16 (\ln (\sqrt[3]{x}) x - \int \frac{x}{3 x} d x)$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Cancel the common factor.

$16 (\ln (\sqrt[3]{x}) x - \int \frac{x}{3 x} d x)$

Step 3.2.2

Rewrite the expression.

$16 (\ln (\sqrt[3]{x}) x - \int \frac{1}{3} d x)$

Step 4

Apply the constant rule.

$16 (\ln (\sqrt[3]{x}) x - (\frac{1}{3} x + C))$

Step 5

Simplify the answer.

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Step 5.1

Rewrite $16 (\ln (\sqrt[3]{x}) x - (\frac{1}{3} x + C))$ as $16 (\ln (\sqrt[3]{x}) x - \frac{1}{3} x) + C$ .

$16 (\ln (\sqrt[3]{x}) x - \frac{1}{3} x) + C$

Step 5.2

Simplify.

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Step 5.2.1

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

$16 (\ln (\sqrt[3]{x}) x - \frac{x}{3}) + C$

Step 5.2.2

To write $\ln (\sqrt[3]{x}) x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$16 (\ln (\sqrt[3]{x}) x \cdot \frac{3}{3} - \frac{x}{3}) + C$

Step 5.2.3

Combine $\ln (\sqrt[3]{x}) x$ and $\frac{3}{3}$ .

$16 (\frac{\ln (\sqrt[3]{x}) x \cdot 3}{3} - \frac{x}{3}) + C$

Step 5.2.4

Combine the numerators over the common denominator.

$16 \frac{\ln (\sqrt[3]{x}) x \cdot 3 - x}{3} + C$

Step 5.2.5

Move $3$ to the left of $\ln (\sqrt[3]{x}) x$ .

$16 \frac{3 \cdot (\ln (\sqrt[3]{x}) x) - x}{3} + C$

Step 5.2.6

Combine $16$ and $\frac{3 \ln (\sqrt[3]{x}) x - x}{3}$ .

$\frac{16 (3 \ln (\sqrt[3]{x}) x - x)}{3} + C$

Step 5.3

Reorder terms.

$\frac{16}{3} (3 \ln (\sqrt[3]{x}) x - x) + C$