Calculus Examples

$\int 8 x^{2} \ln (x) d x$

Step 1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int x^{2} \ln (x) d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{2}$ .

$8 (\ln (x) (\frac{1}{3} x^{3}) - \int \frac{1}{3} x^{3} \frac{1}{x} d x)$

Step 3

Simplify.

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Step 3.1

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

$8 (\ln (x) \frac{x^{3}}{3} - \int \frac{1}{3} x^{3} \frac{1}{x} d x)$

Step 3.2

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

$8 (\frac{\ln (x) x^{3}}{3} - \int \frac{1}{3} x^{3} \frac{1}{x} d x)$

Step 3.3

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

$8 (\frac{\ln (x) x^{3}}{3} - \int \frac{x^{3}}{3} \cdot \frac{1}{x} d x)$

Step 3.4

Multiply $\frac{x^{3}}{3}$ by $\frac{1}{x}$ .

$8 (\frac{\ln (x) x^{3}}{3} - \int \frac{x^{3}}{3 x} d x)$

Step 3.5

Cancel the common factor of $x^{3}$ and $x$ .

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Step 3.5.1

Factor $x$ out of $x^{3}$ .

$8 (\frac{\ln (x) x^{3}}{3} - \int \frac{x \cdot x^{2}}{3 x} d x)$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $x$ out of $3 x$ .

$8 (\frac{\ln (x) x^{3}}{3} - \int \frac{x \cdot x^{2}}{x \cdot 3} d x)$

Step 3.5.2.2

Cancel the common factor.

$8 (\frac{\ln (x) x^{3}}{3} - \int \frac{x \cdot x^{2}}{x \cdot 3} d x)$

Step 3.5.2.3

Rewrite the expression.

$8 (\frac{\ln (x) x^{3}}{3} - \int \frac{x^{2}}{3} d x)$

Step 4

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$8 (\frac{\ln (x) x^{3}}{3} - (\frac{1}{3} \int x^{2} d x))$

Step 5

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$8 (\frac{\ln (x) x^{3}}{3} - \frac{1}{3} (\frac{1}{3} x^{3} + C))$

Step 6

Simplify the answer.

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Step 6.1

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

$8 (\frac{\ln (x) x^{3}}{3} - \frac{1}{3} (\frac{x^{3}}{3} + C))$

Step 6.2

Rewrite $8 (\frac{\ln (x) x^{3}}{3} - \frac{1}{3} (\frac{x^{3}}{3} + C))$ as $8 (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) + C$ .

$8 (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) + C$

Step 6.3

Reorder terms.

$8 (\frac{1}{3} \ln (x) x^{3} - \frac{1}{9} x^{3}) + C$