Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Multiply $3$ by $- 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Multiply $2$ by $- 1$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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