Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x^{2})$ and $d v = x^{3}$ .

$\ln (x^{2}) (\frac{1}{4} x^{4}) - \int \frac{1}{4} x^{4} \frac{2}{x} d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$\ln (x^{2}) \frac{x^{4}}{4} - \int \frac{1}{4} x^{4} \frac{2}{x} d x$

Step 2.2

Combine $\ln (x^{2})$ and $\frac{x^{4}}{4}$ .

$\frac{\ln (x^{2}) x^{4}}{4} - \int \frac{1}{4} x^{4} \frac{2}{x} d x$

Step 3

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{\ln (x^{2}) x^{4}}{4} - (\frac{1}{4} \int x^{4} \frac{2}{x} d x)$

Step 4

Simplify.

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Step 4.1

Combine $x^{4}$ and $\frac{2}{x}$ .

$\frac{\ln (x^{2}) x^{4}}{4} - (\frac{1}{4} \int \frac{x^{4} \cdot 2}{x} d x)$

Step 4.2

Move $2$ to the left of $x^{4}$ .

$\frac{\ln (x^{2}) x^{4}}{4} - (\frac{1}{4} \int \frac{2 \cdot x^{4}}{x} d x)$

Step 4.3

Cancel the common factor of $x^{4}$ and $x$ .

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Step 4.3.1

Factor $x$ out of $2 x^{4}$ .

$\frac{\ln (x^{2}) x^{4}}{4} - (\frac{1}{4} \int \frac{x (2 x^{3})}{x} d x)$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Raise $x$ to the power of $1$ .

$\frac{\ln (x^{2}) x^{4}}{4} - (\frac{1}{4} \int \frac{x (2 x^{3})}{x^{1}} d x)$

Step 4.3.2.2

Factor $x$ out of $x^{1}$ .

$\frac{\ln (x^{2}) x^{4}}{4} - (\frac{1}{4} \int \frac{x (2 x^{3})}{x \cdot 1} d x)$

Step 4.3.2.3

Cancel the common factor.

$\frac{\ln (x^{2}) x^{4}}{4} - (\frac{1}{4} \int \frac{x (2 x^{3})}{x \cdot 1} d x)$

Step 4.3.2.4

Rewrite the expression.

$\frac{\ln (x^{2}) x^{4}}{4} - (\frac{1}{4} \int \frac{2 x^{3}}{1} d x)$

Step 4.3.2.5

Divide $2 x^{3}$ by $1$ .

$\frac{\ln (x^{2}) x^{4}}{4} - (\frac{1}{4} \int 2 x^{3} d x)$

$\frac{\ln (x^{2}) x^{4}}{4} - \frac{1}{4} \int 2 x^{3} d x$

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\frac{\ln (x^{2}) x^{4}}{4} - \frac{1}{4} (2 \int x^{3} d x)$

Step 6

Simplify.

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Step 6.1

Multiply $2$ by $- 1$ .

$\frac{\ln (x^{2}) x^{4}}{4} - 2 (\frac{1}{4}) \int x^{3} d x$

Step 6.2

Combine $- 2$ and $\frac{1}{4}$ .

$\frac{\ln (x^{2}) x^{4}}{4} + \frac{- 2}{4} \int x^{3} d x$

Step 6.3

Cancel the common factor of $- 2$ and $4$ .

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Step 6.3.1

Factor $2$ out of $- 2$ .

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

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $2$ out of $4$ .

$\frac{\ln (x^{2}) x^{4}}{4} + \frac{2 \cdot - 1}{2 \cdot 2} \int x^{3} d x$

Step 6.3.2.2

Cancel the common factor.

$\frac{\ln (x^{2}) x^{4}}{4} + \frac{2 \cdot - 1}{2 \cdot 2} \int x^{3} d x$

Step 6.3.2.3

Rewrite the expression.

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

Step 6.4

Move the negative in front of the fraction.

$\frac{\ln (x^{2}) x^{4}}{4} - \frac{1}{2} \int x^{3} d x$

Step 7

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 8

Simplify the answer.

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Step 8.1

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

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

Step 8.2

Simplify.

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Step 8.2.1

Combine $\frac{1}{4}$ and $x^{4}$ .

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

Step 8.2.2

Multiply $\frac{x^{4}}{4}$ by $\frac{1}{2}$ .

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

Step 8.2.3

Multiply $4$ by $2$ .

$\frac{\ln (x) x^{4}}{2} - \frac{x^{4}}{8} + C$

Step 8.3

Reorder terms.

$\frac{1}{2} \ln (x) x^{4} - \frac{1}{8} x^{4} + C$