How do you find the antiderivative of f(x)=8x^3+5x^2-9x+3?

Apr 5, 2018

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Explanation:

The anti-derivative or primitive function is achieved by integrating the function.

A rule of thumb here is if asked to find the antiderivative/integral of a function which is polynomial:
Take the function and increase all indices of $x$ by 1, and then divide each term by their new index of $x$.

Or mathematically:

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

You also add a constant to the function, although the constant will be arbitrary in this problem.

Now, using our rule we can find the primitive function, $F \left(x\right)$.

$F \left(x\right) = \left(\frac{8 {x}^{3 + 1}}{3 + 1}\right) + \left(\frac{5 {x}^{2 + 1}}{2 + 1}\right) + \left(\frac{- 9 {x}^{1 + 1}}{1 + 1}\right) + \left(\frac{3 {x}^{0 + 1}}{0 + 1}\right) \left(+ C\right)$

If the term in question does not include an x, it will have an x in the primitive function because:

${x}^{0} = 1$ So raising the index of all $x$ terms turns ${x}^{0}$ to ${x}^{1}$ which is equal to $x$.

So , simplified the antiderivative becomes:

$F \left(x\right) = 2 {x}^{4} + \left(\frac{5 {x}^{3}}{3}\right) - \left(\frac{9 {x}^{2}}{2}\right) + 3 x \left(+ C\right)$

Apr 5, 2018

$2 {x}^{4} + \frac{5}{3} {x}^{3} - \frac{9}{2} {x}^{2} + 3 x + C$

Explanation:

The anti-derivative of a function $f \left(x\right)$ is given by $F \left(x\right)$, where $F \left(x\right) = \int f \left(x\right) \setminus \mathrm{dx}$. You can think of the anti-derivative as the integral of the function.

Therefore,

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

$= \int 8 {x}^{3} + 5 {x}^{2} - 9 x + 3$

We are going to need some integral rules to solve this problem. They are:

$\int {a}^{x} \setminus \mathrm{dx} = \frac{{a}^{x + 1}}{x + 1} + C$

$\int a \setminus \mathrm{dx} = a x + C$

$\int \left(f \left(x\right) + g \left(x\right)\right) \setminus \mathrm{dx} = \int f \left(x\right) \setminus \mathrm{dx} + \int g \left(x\right) \setminus \mathrm{dx}$

And so, we get:

$\textcolor{b l u e}{= \overline{\underline{| 2 {x}^{4} + \frac{5}{3} {x}^{3} - \frac{9}{2} {x}^{2} + 3 x + C |}}}$