Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

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

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

Step 4.2.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

Cancel the common factor.

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

Step 4.2.2.4

Rewrite the expression.

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

Step 4.2.2.5

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

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

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

Step 5

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

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

Step 6

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

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

Step 7

The natural logarithm of zero is undefined.

Undefined