Evaluate the Integral integral of (5-3z-2z^3) with respect to z

$\int (5 - 3 z - 2 z^{3}) d z$

Step 1

Remove parentheses.

$\int 5 - 3 z - 2 z^{3} d z$

Step 2

Split the single integral into multiple integrals.

$\int 5 d z + \int - 3 z d z + \int - 2 z^{3} d z$

Step 3

Apply the constant rule.

$5 z + C + \int - 3 z d z + \int - 2 z^{3} d z$

Step 4

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

$5 z + C - 3 \int z d z + \int - 2 z^{3} d z$

Step 5

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

$5 z + C - 3 (\frac{1}{2} z^{2} + C) + \int - 2 z^{3} d z$

Step 6

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

$5 z + C - 3 (\frac{1}{2} z^{2} + C) - 2 \int z^{3} d z$

Step 7

By the Power Rule, the integral of $z^{3}$ with respect to $z$ is $\frac{1}{4} z^{4}$ .

$5 z + C - 3 (\frac{1}{2} z^{2} + C) - 2 (\frac{1}{4} z^{4} + C)$

Step 8

Simplify.

$5 z - \frac{3 z^{2}}{2} - 2 (\frac{1}{4} z^{4}) + C$

Simplify.

Combine $\frac{1}{4}$ and $z^{4}$ .

$5 z - \frac{3 z^{2}}{2} - 2 \frac{z^{4}}{4} + C$

Combine $- 2$ and $\frac{z^{4}}{4}$ .

$5 z - \frac{3 z^{2}}{2} + \frac{- 2 z^{4}}{4} + C$

Cancel the common factor of $- 2$ and $4$ .

Factor $2$ out of $- 2 z^{4}$ .

$5 z - \frac{3 z^{2}}{2} + \frac{2 (- z^{4})}{4} + C$

Cancel the common factors.

Factor $2$ out of $4$ .

$5 z - \frac{3 z^{2}}{2} + \frac{2 (- z^{4})}{2 (2)} + C$

Cancel the common factor.

$5 z - \frac{3 z^{2}}{2} + \frac{2 (- z^{4})}{2 \cdot 2} + C$

Rewrite the expression.

$5 z - \frac{3 z^{2}}{2} + \frac{- z^{4}}{2} + C$

Move the negative in front of the fraction.

$5 z - \frac{3 z^{2}}{2} - \frac{z^{4}}{2} + C$

Step 9

Reorder terms.

$5 z - \frac{3}{2} z^{2} - \frac{1}{2} z^{4} + C$