Calculus Examples

$p (z) = {(\sqrt{z})}^{7}$

Step 1

The function $P (z)$ can be found by finding the indefinite integral of the derivative $p (z)$ .

$P (z) = \int p (z) d z$

Step 2

Set up the integral to solve.

$P (z) = \int {(\sqrt{z})}^{7} d z$

Step 3

Rewrite ${(\sqrt{z})}^{7}$ as $\sqrt{z^{7}}$ .

$\int \sqrt{z^{7}} d z$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z^{7}}$ as $z^{\frac{7}{2}}$ .

$\int z^{\frac{7}{2}} d z$

Step 5

By the Power Rule, the integral of $z^{\frac{7}{2}}$ with respect to $z$ is $\frac{2}{9} z^{\frac{9}{2}}$ .

$\frac{2}{9} z^{\frac{9}{2}} + C$

Step 6

The answer is the antiderivative of the function $p (z) = {(\sqrt{z})}^{7}$ .

$P (z) =$ $\frac{2}{9} z^{\frac{9}{2}} + C$