# Integrals of Polynomial functions

## Key Questions

• Because this equation only consists of terms added together, you can integrate them separately and add the results, giving us:

$\int {x}^{3} + 4 {x}^{2} + 5 \mathrm{dx} = \int {x}^{3} \mathrm{dx} + \int 4 {x}^{2} \mathrm{dx} + \int 5 \mathrm{dx}$

Each of these terms can be integrated using the Power Rule for integration, which is:

$\int {x}^{n} \mathrm{dx} = {x}^{n + 1} / \left(n + 1\right) + C$

Plugging our 3 terms into this formula, we have:

$\int {x}^{3} \mathrm{dx} = {x}^{3 + 1} / \left(3 + 1\right) = {x}^{4} / 4$

$\int 4 {x}^{2} \mathrm{dx} = \frac{4 {x}^{2 + 1}}{2 + 1} = \frac{4 {x}^{3}}{3}$

$\int 5 \mathrm{dx} = \int 5 {x}^{0} \mathrm{dx} = \frac{5 {x}^{0 + 1}}{0 + 1} = \frac{5 {x}^{1}}{1} = 5 x$

Now we arrive at our final answer by adding these together, remembering to add our constant ($C$) on the end:

$\int {x}^{3} + 4 {x}^{2} + 5 \mathrm{dx} = {x}^{4} / 4 + \frac{4 {x}^{3}}{3} + 5 x + C$

• First you integrate the function:

$\int {x}^{3} + 2 {x}^{2} - 8 x - 1 = \frac{1}{4} {x}^{4} + \frac{2}{3} {x}^{3} - 4 {x}^{2} - x$

$\frac{1}{4} {\left(4\right)}^{4} + \frac{2}{3} {\left(4\right)}^{3} - 4 {\left(4\right)}^{2} - 4 = \frac{1}{4} \left(256\right) + \frac{2}{3} \left(64\right) - 4 \left(16\right) - 4$

Solving that out yields:

$64 + \frac{128}{3} - 64 - 4 = \frac{116}{3} \left(\mathmr{and} 38.66666\right)$

Next you would substitute in 0, but looking at the equation, you can see that subbing 0 in will just yield zero. So last you do $\frac{116}{3} - 0$, which of course is just $\frac{116}{3}$, and that's your answer.

• Let

$f \left(x\right) = {a}_{n} {x}^{n} + {a}_{n - 1} {x}^{n - 1} + \cdots + {a}_{1} x + {a}_{0}$.

An antiderivative $F \left(x\right)$of $f \left(x\right)$ can be found by

$F \left(x\right) = \int f \left(x\right) \mathrm{dx}$

$= \int \left({a}_{n} {x}^{n} + {a}_{n - 1} {x}^{n - 1} + \cdots + {a}_{1} x + {a}_{0}\right) \mathrm{dx}$

$= {a}_{n} / \left\{n + 1\right\} {x}^{n + 1} + {a}_{n - 1} / n {x}^{n} + \cdots + {a}_{1} / 2 {x}^{2} + {a}_{0} x + C$.

I hope that this was helpful.