Calculus Examples

$\int 13 z^{3} e^{z} d z$

Step 1

Since $13$ is constant with respect to $z$ , move $13$ out of the integral.

$13 \int z^{3} e^{z} d z$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = z^{3}$ and $d v = e^{z}$ .

$13 (z^{3} e^{z} - \int e^{z} (3 z^{2}) d z)$

Step 3

Since $3$ is constant with respect to $z$ , move $3$ out of the integral.

$13 (z^{3} e^{z} - (3 \int e^{z} (z^{2}) d z))$

Step 4

Multiply $3$ by $- 1$ .

$13 (z^{3} e^{z} - 3 \int e^{z} (z^{2}) d z)$

Step 5

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

$13 (z^{3} e^{z} - 3 (z^{2} e^{z} - \int e^{z} (2 z) d z))$

Step 6

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

$13 (z^{3} e^{z} - 3 (z^{2} e^{z} - (2 \int e^{z} (z) d z)))$

Step 7

Multiply $2$ by $- 1$ .

$13 (z^{3} e^{z} - 3 (z^{2} e^{z} - 2 \int e^{z} (z) d z))$

Step 8

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

$13 (z^{3} e^{z} - 3 (z^{2} e^{z} - 2 (z e^{z} - \int e^{z} d z)))$

Step 9

The integral of $e^{z}$ with respect to $z$ is $e^{z}$ .

$13 (z^{3} e^{z} - 3 (z^{2} e^{z} - 2 (z e^{z} - (e^{z} + C))))$

Step 10

Rewrite $13 (z^{3} e^{z} - 3 (z^{2} e^{z} - 2 (z e^{z} - (e^{z} + C))))$ as $13 (z^{3} e^{z} - 3 z^{2} e^{z} + 6 z e^{z} - 6 e^{z}) + C$ .

$13 (z^{3} e^{z} - 3 z^{2} e^{z} + 6 z e^{z} - 6 e^{z}) + C$