# What is the antiderivative of xsqrtx?

##### 2 Answers
Jul 27, 2015

You can simply multiply them together (more explicitly).

$x \sqrt{x} = {x}^{\text{3/2}}$

And then just use the reverse Power Rule.

d/(dx)[x^("3/2")] = 2/5x^("5/2")

Then, since an antiderivative is a generalization of what an integral does, they are almost the same thing. Therefore, we add a constant to imply that you get every single function that is within the antiderivative's slope field.

(notice the various vertical-shift variations of a single function forms the slope field)

$\to \textcolor{b l u e}{\frac{2}{5} {x}^{\text{5/2}} + C}$

Jul 27, 2015

$\frac{2}{5} {x}^{\frac{5}{2}}$

#### Explanation:

Note : $x \sqrt{x} = x {x}^{\frac{1}{2}} = {x}^{1 + \frac{1}{2}} = {x}^{\frac{3}{2}}$
Therefore, $\int x \sqrt{x} \mathrm{dx} = \int {x}^{\frac{3}{2}} \mathrm{dx} = \frac{{x}^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} = \frac{2}{5} {x}^{\frac{5}{2}}$