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Thread: Can someone help me understand this defintie integral?

  1. #1
    ORT
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    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 help me understand this defintie integral?-win_20190515_11_08_51_pro.jpg
    Can someone please help me and explain to me step by step what I have to do?

    Thank you very much in advance!
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: Can someone help me understand this defintie integral?

    Quote Originally Posted by ORT View Post
    Hello everyone

    I'm having a difficult time to understand definite integrals.
    I have the following integral: Click image for larger version. 

Name:	WIN_20190515_11_08_51_Pro.jpg 
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ID:	39378
    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
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    Re: Can someone help me understand this defintie integral?

    first step

    substitution $\displaystyle u=\sqrt{x}$
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    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.
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  5. #5
    ORT
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    Re: Can someone help me understand this defintie integral?

    Where do you get the 2 from?
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    Re: Can someone help me understand this defintie integral?

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

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

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