Calculus Examples

$x \cdot \ln (x)$

Step 1

Write $x \cdot \ln (x)$ as a function.

$f (x) = x \cdot \ln (x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x \cdot \ln (x) d x$

Step 4

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

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

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

Step 6

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

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

Step 7

Simplify.

Tap for more steps...

Step 7.1

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

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

Step 7.2

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

Tap for more steps...

Step 7.2.1

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

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

Step 7.2.2

Cancel the common factors.

Tap for more steps...

Step 7.2.2.1

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

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

Step 7.2.2.2

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

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

Step 7.2.2.3

Cancel the common factor.

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

Step 7.2.2.4

Rewrite the expression.

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

Step 7.2.2.5

Divide $x$ by $1$ .

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

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

Step 8

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

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

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

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

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

Step 9.2

Simplify.

Tap for more steps...

Step 9.2.1

Combine $\frac{1}{2}$ and $\ln (x)$ .

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

Step 9.2.2

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

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

Step 9.2.3

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

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

Step 9.2.4

Multiply $2$ by $2$ .

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

Step 9.3

Combine $x^{2}$ and $\frac{1}{4}$ .

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

Step 9.4

Reorder terms.

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

Step 10

The answer is the antiderivative of the function $f (x) = x \cdot \ln (x)$ .

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