Calculus Examples

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

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

Step 1

Since

8

$8$ is constant with respect to

x

$x$ , move

8

$8$ out of the integral.

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

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

Step 2

Integrate by parts using the formula

\int u d v = u v - \int v d u

$\int u d v = u v - \int v d u$ , where

u = ln (x)

$u = \ln (x)$ and

d v = x^{2}

$d v = x^{2}$ .

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

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

Step 3

Simplify.

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Step 3.1

Combine

\frac{1}{3}

$\frac{1}{3}$ and

x^{3}

$x^{3}$ .

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

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

Step 3.2

Combine

ln (x)

$\ln (x)$ and

\frac{x^{3}}{3}

$\frac{x^{3}}{3}$ .

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

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

Step 3.3

Combine

\frac{1}{3}

$\frac{1}{3}$ and

x^{3}

$x^{3}$ .

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

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

Step 3.4

Multiply

\frac{x^{3}}{3}

$\frac{x^{3}}{3}$ by

\frac{1}{x}

$\frac{1}{x}$ .

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

$x^{3}$ and

x

$x$ .

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Step 3.5.1

Factor

x

$x$ out of

x^{3}

$x^{3}$ .

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

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

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor

x

$x$ out of

3 x

$3 x$ .

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

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