I can't figure out why when integrating
$\displaystyle
-4{x}^\frac{3}{2}
$
you get
$\displaystyle
-\displaystyle{\frac{8}{3}{x}}^\frac{3}{2}
$
and not
$\displaystyle
-8{x}^\frac{3}{2}
$
Why does it become a fraction?
You shouldn't get either of those answers you've listed...
If $\displaystyle f(x) = ax^n$ then it's antiderivative is $\displaystyle F(x) = \int{ax^n\,dx} = \frac{ax^{n + 1}}{n + 1}+C$.
In other words, you add 1 to your power, and then divide by the new power.
In your case you have $\displaystyle -4x^{\frac{3}{2}}$.
Add 1 to the power and you get $\displaystyle \frac{5}{2}$.
Divide by your new power and you get
$\displaystyle -4\div\frac{5}{2}$
$\displaystyle = -4\times\frac{2}{5}$
$\displaystyle = -\frac{8}{5}$.
So your antiderivative is $\displaystyle -\frac{8x^{\frac{5}{2}}}{5} + C$ where C is a constant.
Thanks, I forgot to flip the fraction and then times it
Totally correct, I wrote down the wrong quesion as $\displaystyle -4{x}^\frac{3}{2}$ which was my partially intergrated answer using $\displaystyle {ax^{n + 1}}$, the actual question was $\displaystyle -4{x}^\frac{1}{2}$!