Calculus Examples

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

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

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

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

Step 5.2.2.2

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

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

Step 5.2.2.3

Cancel the common factor.

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

Step 5.2.2.4

Rewrite the expression.

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

Step 5.2.2.5

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

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

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

Simplify.

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Step 7.2.1

Combine $\frac{1}{4}$ and $\ln (x)$ .

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

Step 7.2.2

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

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

Step 7.2.3

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

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

Step 7.2.4

Multiply $4$ by $4$ .

$\frac{\ln (x) x^{4}}{4} - \frac{1}{16} x^{4} + C$

$\frac{1}{4} \ln (x) x^{4} - \frac{1}{16} x^{4} + C$