Question

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

Answer 1

Like this :

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)/(n+1)(+C)`

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(x)`.

`F(x)=((8x^(3+1))/(3+1))+((5x^(2+1))/(2+1))+((-9x^(1+1))/(1+1))+((3x^(0+1))/(0+1))(+C)`

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(x)=2x^4+((5x^3)/3)-((9x^2)/2)+3x(+C)`

Answer 2

`2x^4+5/3x^3-9/2x^2+3x+C`

Explanation:

The anti-derivative of a function `f(x)` is given by `F(x)`, where `F(x)=intf(x) \ dx`. You can think of the anti-derivative as the integral of the function.

Therefore,

`F(x)=intf(x) \ dx`

`=int8x^3+5x^2-9x+3`

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

`inta^x \ dx=(a^(x+1))/(x+1)+C`

`inta \ dx=ax+C`

`int(f(x)+g(x)) \ dx=intf(x) \ dx+intg(x) \ dx`

And so, we get:

`color(blue)(=barul(|2x^4+5/3x^3-9/2x^2+3x+C|))`