Calculus Examples

$\int_{0}^{2} x^{2} \cdot \ln (x) d x$

Step 1

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

$\ln (x) (\frac{1}{3} x^{3})]_{0}^{2} - \int_{0}^{2} \frac{1}{3} x^{3} \frac{1}{x} d x$

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

$\ln (x) \frac{x^{3}}{3}]_{0}^{2} - \int_{0}^{2} \frac{1}{3} x^{3} \frac{1}{x} d x$

Step 2.2

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

$\frac{\ln (x) x^{3}}{3}]_{0}^{2} - \int_{0}^{2} \frac{1}{3} x^{3} \frac{1}{x} d x$

Step 3

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

$\frac{\ln (x) x^{3}}{3}]_{0}^{2} - (\frac{1}{3} \int_{0}^{2} x^{3} \frac{1}{x} d x)$

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

$\frac{\ln (x) x^{3}}{3}]_{0}^{2} - (\frac{1}{3} \int_{0}^{2} \frac{x^{3}}{x} d x)$

Step 4.2

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

Tap for more steps...

Step 4.2.1

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

$\frac{\ln (x) x^{3}}{3}]_{0}^{2} - (\frac{1}{3} \int_{0}^{2} \frac{x \cdot x^{2}}{x} d x)$

Step 4.2.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.1

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

$\frac{\ln (x) x^{3}}{3}]_{0}^{2} - (\frac{1}{3} \int_{0}^{2} \frac{x \cdot x^{2}}{x^{1}} d x)$

Step 4.2.2.2

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

$\frac{\ln (x) x^{3}}{3}]_{0}^{2} - (\frac{1}{3} \int_{0}^{2} \frac{x \cdot x^{2}}{x \cdot 1} d x)$

Step 4.2.2.3

Cancel the common factor.

$\frac{\ln (x) x^{3}}{3}]_{0}^{2} - (\frac{1}{3} \int_{0}^{2} \frac{x \cdot x^{2}}{x \cdot 1} d x)$

Step 4.2.2.4

Rewrite the expression.

$\frac{\ln (x) x^{3}}{3}]_{0}^{2} - (\frac{1}{3} \int_{0}^{2} \frac{x^{2}}{1} d x)$

Step 4.2.2.5

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

$\frac{\ln (x) x^{3}}{3}]_{0}^{2} - (\frac{1}{3} \int_{0}^{2} x^{2} d x)$

$\frac{\ln (x) x^{3}}{3}]_{0}^{2} - \frac{1}{3} \int_{0}^{2} x^{2} d x$

Step 5

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

$\frac{\ln (x) x^{3}}{3}]_{0}^{2} - \frac{1}{3} \frac{1}{3} x^{3}]_{0}^{2}$

Step 6

Evaluate $\frac{\ln (x) x^{3}}{3}$ at $2$ and at $0$ .

$(\frac{\ln (2) \cdot 2^{3}}{3}) - \frac{\ln (0) x^{3}}{3} - \frac{1}{3} \frac{1}{3} x^{3}]_{0}^{2}$

Step 7

The natural logarithm of zero is undefined.

Undefined