# Thread: Can someone help me understand this defintie integral?

1. ## Can someone help me understand this defintie integral?

Hello everyone

I'm having a difficult time to understand definite integrals.
I have the following integral:
Can someone please help me and explain to me step by step what I have to do?

Thank you very much in advance!

2. ## Re: Can someone help me understand this defintie integral?

Originally Posted by ORT
Hello everyone

I'm having a difficult time to understand definite integrals.
I have the following integral:
Can someone please help me and explain to me step by step what I have to do?

Thank you very much in advance!
The first thing to do is see about getting rid of that square root. Since x has to be positive anyway I would suggest the substitution $\displaystyle x = y^2$. Then $\displaystyle dx = 2y~dy$ and your integral now becomes:
$\displaystyle \int_0^4 \dfrac{1}{1 + \sqrt{x}}~dx = \int_0^2 \dfrac{1}{1 + y}~2y~dy$

which is a much more familiar form. Can you take it from here?

-Dan

3. ## Re: Can someone help me understand this defintie integral?

first step

substitution $\displaystyle u=\sqrt{x}$

4. ## Re: Can someone help me understand this defintie integral?

Here's another way:

Let $\displaystyle \ u = 1 + \sqrt{x}$

As the limits of integration for x are from 0 to 4, then for u, they are from 1 to 3.

$\displaystyle u = 1 + \sqrt{x}$

$\displaystyle u - 1 = \sqrt{x}$

$\displaystyle (u - 1)^2 = x$

$\displaystyle dx = 2(u - 1)du$

The integrand will be $\displaystyle \ \ \dfrac{2(u - 1)du}{u} \ =$

$\displaystyle 2\bigg(1 - \dfrac{1}{u}\bigg)du$.

And then you can continue.

5. ## Re: Can someone help me understand this defintie integral?

Where do you get the 2 from?

6. ## Re: Can someone help me understand this defintie integral?

Originally Posted by ORT
Where do you get the 2 from?
$\dfrac{d}{dx}\left[(u-1)^2 = x\right]$

$2(u-1) \cdot \dfrac{du}{dx} = 1$

$2(u-1) \, du = dx$