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Math Help - Integrate e^x(1-e^2x)^(1/2)dx

  1. #1
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    Integrate e^x(1-e^2x)^(1/2)dx

    Hi, I was wondering if you guys could check my work for indefinite integral of (e^x)(sqrt{1-e^2x)

    Let u=e^x

    The integral becomes:

    integral of sqrt(1-u^2)

    Then I do trigonometric substitution where I let u=sinx, and I get:

    1/2sin^-1(x) + (x/2)*(1 - x^2)^1/2 + c.

    However, the answer in my textbook is -1/3(1-e^(2x))^(3/2).

    I was just wondering what I did wrong for the question.

    If my complete process is wrong, what should I have done to solve the question?

    Thanks in advance
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  2. #2
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    Re: Integrate e^x(1-e^2x)^(1/2)dx

    Hi !
    there is a mishmash in your answer because the same symbol x is used for two different variables :
    u=e^x
    u=sinx
    leading to e^x=sinx which is false.
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  3. #3
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    Re: Integrate e^x(1-e^2x)^(1/2)dx

    You are just right put e^x=u that gives e^x dx = du
    the question becomes integral of (1-u^2)^1/2 du and that is a standard for for which the integral is given by
    (a^2-x^2)^1/2 = x/2 (a^2-x^2)^1/2 + 1/2 a^2 sin^-1 (x/a) + C
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  4. #4
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    Re: Integrate e^x(1-e^2x)^(1/2)dx

    Hello, verasi!

    Your substitution is faulty.


    \int e^x \sqrt{1-e^{2x}}\,dx

    We have: . \int \sqrt{1-e^{2x}}\,(e^x\,dx)

    Let u = e^x \quad\Rightarrow\quad du = e^x\,dx

    Substitute: . \int \sqrt{1-u^2}\,du

    Let u = \sin\theta \quad\Rightarrow\quad du = \cos\theta\,d\theta

    Substitute: . \int \sqrt{1-\sin^2\!\theta}\,\cos\theta\,d\theta \;=\;\int\cos^2\!\theta\,d\theta \;=\;\tfrac{1}{2}\int(1 + \cos2\theta)\,d\theta

    . . . . . . . . =\;\tfrac{1}{2}\left(\theta + \tfrac{1}{2}\sin2\theta\right) + C \;=\;\tfrac{1}{2}(\theta + \sin\theta\cos\theta) + C


    Back-substitute: . \tfrac{1}{2}\left(\arcsin u + u\sqrt{1-u^2}\right) + C


    Back-substitute: . \tfrac{1}{2}\left(\arcsin e^x + e^x\sqrt{1-e^{2x}}\right) + C
    Thanks from topsquark
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    Re: Integrate e^x(1-e^2x)^(1/2)dx

    Thanks for the reply, I guess my initial answer was right then. Apparently the textbook has a different answer. It's -1/3(1-e^(2x))^(3/2).

    I'm guessing the textbook used a different method then (integrate by part maybe? I didn't try it seemed way too complicated).

    T.T wasted 30min on this question trying to figure out what I did wrong
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  6. #6
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    Re: Integrate e^x(1-e^2x)^(1/2)dx

    Quote Originally Posted by verasi View Post
    Thanks for the reply, I guess my initial answer was right then. Apparently the textbook has a different answer. It's -1/3(1-e^(2x))^(3/2).

    I'm guessing the textbook used a different method then (integrate by part maybe? I didn't try it seemed way too complicated).

    T.T wasted 30min on this question trying to figure out what I did wrong
    Hi!

    Fortunately, any different method leads to the same result ! The results are different if there is a mistake somewhere (in calculus or if the integrals are not the same) See attachment :
    Attached Thumbnails Attached Thumbnails Integrate e^x(1-e^2x)^(1/2)dx-integrals.jpg  
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