# Susie's question at Yahoo! Answers regarding minimizing a definite integral

#### MarkFL

Staff member
Here is the question:

Find the value of a >0 that minimizes Int_{a}^{a^2} dx/(x_sqrt(x))

I am asking for what value of a the integral from a to a^2 dx/(x*sqrt(x)) will generate the smallest number.

I'm sorry, I meant to type:

The integral from a to a^2 of dx/(x+sqrt(x))
Here is a link to the question:

I have posted a link there to this topic so the OP can find my response.

#### MarkFL

Staff member
Re: Susie's question at Yahoo! Answers regarding minimzing a definite integral

Hello Susie,

We are given to minimize:

$$\displaystyle g(a)=\int_a^{a^2}\frac{dx}{x+\sqrt{x}}$$

Obviously, we want to equate the derivative to zero and find the critical value(s). So we may utilize the anti-derivative form of the FTOC, and differentiate using the chain rule:

$$\displaystyle \frac{d}{da}\left(\int_a^{a^2}\frac{dx}{x+\sqrt{x}} \right)=0$$

$$\displaystyle \frac{1}{a^2+a}\cdot2a-\frac{1}{a+\sqrt{a}}=0$$

$$\displaystyle \frac{2}{a+1}=\frac{1}{a+\sqrt{a}}$$

$$\displaystyle a+2\sqrt{a}-1=0$$

$$\displaystyle \sqrt{a}=\sqrt{2}-1$$
$$\displaystyle a=3-2\sqrt{2}$$