Calculus Examples

$\int 12 x \ln (3 x) d x$

Step 1

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int x \ln (3 x) d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (3 x)$ and $d v = x$ .

$12 (\ln (3 x) (\frac{1}{2} x^{2}) - \int \frac{1}{2} x^{2} \frac{1}{x} d x)$

Step 3

Simplify.

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Combine $\frac{1}{2}$ and $x^{2}$ .

$12 (\ln (3 x) \frac{x^{2}}{2} - \int \frac{1}{2} x^{2} \frac{1}{x} d x)$

Combine $\ln (3 x)$ and $\frac{x^{2}}{2}$ .

$12 (\frac{\ln (3 x) x^{2}}{2} - \int \frac{1}{2} x^{2} \frac{1}{x} d x)$

Combine $\frac{1}{2}$ and $x^{2}$ .

$12 (\frac{\ln (3 x) x^{2}}{2} - \int \frac{x^{2}}{2} \cdot \frac{1}{x} d x)$

Multiply $\frac{x^{2}}{2}$ by $\frac{1}{x}$ .

$12 (\frac{\ln (3 x) x^{2}}{2} - \int \frac{x^{2}}{2 x} d x)$

Cancel the common factor of $x^{2}$ and $x$ .

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Factor $x$ out of $x^{2}$ .

$12 (\frac{\ln (3 x) x^{2}}{2} - \int \frac{x \cdot x}{2 x} d x)$

Cancel the common factors.

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Factor $x$ out of $2 x$ .

$12 (\frac{\ln (3 x) x^{2}}{2} - \int \frac{x \cdot x}{x \cdot 2} d x)$

Cancel the common factor.

$12 (\frac{\ln (3 x) x^{2}}{2} - \int \frac{x \cdot x}{x \cdot 2} d x)$

Rewrite the expression.

$12 (\frac{\ln (3 x) x^{2}}{2} - \int \frac{x}{2} d x)$

Step 4

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$12 (\frac{\ln (3 x) x^{2}}{2} - (\frac{1}{2} \int x d x))$

Step 5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$12 (\frac{\ln (3 x) x^{2}}{2} - \frac{1}{2} (\frac{1}{2} x^{2} + C))$

Step 6

Simplify the answer.

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Combine $\frac{1}{2}$ and $x^{2}$ .

$12 (\frac{\ln (3 x) x^{2}}{2} - \frac{1}{2} (\frac{x^{2}}{2} + C))$

Rewrite $12 (\frac{\ln (3 x) x^{2}}{2} - \frac{1}{2} (\frac{x^{2}}{2} + C))$ as $12 (\frac{\ln (3 x) x^{2}}{2} - \frac{x^{2}}{4}) + C$ .

$12 (\frac{\ln (3 x) x^{2}}{2} - \frac{x^{2}}{4}) + C$

Reorder terms.

$12 (\frac{1}{2} \ln (3 x) x^{2} - \frac{1}{4} x^{2}) + C$