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Math Help - Integration with Natural Logs

  1. #1
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    Integration with Natural Logs

    I am having issues with some problems with the integration of natural logs/substitution. I was fine at the beginning of the chapter, but now I'm having some problems. These are a few examples of the type I'm stuck on:

    1) The integral of (1/(1+2x^(1/2))dx

    2) The integral of (cot(theta/3))d-theta

    For the first I tried substitution and got u=1+2x^(1/2), but the du is what is confusing me. I'm just not sure how to proceed.

    And the second one I realize you change cot into cos/sin, but I'm confused as to why in the answer a 1/3 (and a 3 to counteract it) is added to the integral? I'm just confused with the process...

    If anyone could kindly guide me in the right direction, I would be forever thankful! Last quiz of the semester is tomorrow morning and I really want to do well on it!
    Last edited by mr fantastic; December 8th 2010 at 02:11 PM. Reason: Deleted begging from title.
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    Quote Originally Posted by Chinnie15 View Post
    1) The integral of (1/(1+2x^(1/2))dx

    Is this \displaystyle\int \frac{1}{\sqrt{1+2x}} ~dx or \displaystyle\int \frac{1}{1+\sqrt{2x}} ~dx ??
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    The second one. Sorry about that.
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    Quote Originally Posted by Chinnie15 View Post
    I am having issues with some problems with the integration of natural logs/substitution. I was fine at the beginning of the chapter, but now I'm having some problems. These are a few examples of the type I'm stuck on:

    1) The integral of (1/(1+2x^(1/2))dx

    2) The integral of (cot(theta/3))d-theta

    For the first I tried substitution and got u=1+2x^(1/2), but the du is what is confusing me. I'm just not sure how to proceed.

    And the second one I realize you change cot into cos/sin, but I'm confused as to why in the answer a 1/3 (and a 3 to counteract it) is added to the integral? I'm just confused with the process...

    If anyone could kindly guide me in the right direction, I would be forever thankful! Last quiz of the semester is tomorrow morning and I really want to do well on it!
    \displaystyle 1)  u = 1 + 2\sqrt{x}<br />

    \displaystyle du = \frac{1}{\sqrt{x}} dx

    \displaystyle dx = \sqrt{x} \, du<br />

    \displaystyle dx = \frac{u-1}{2} \, du

    \displaystyle \int \frac{1}{1+2\sqrt{x}} \, dx

    substitute ...

    \displaystyle \int \frac{1}{u} \cdot \frac{u-1}{2} \, du<br />

    \displaystyle \frac{1}{2} \int 1 - \frac{1}{u} \, du

    integrate and back substitute.


    2) \displaystyle \int \frac{\cos\left(\frac{t}{3}\right)}{\sin\left(\fra  c{t}{3}\right)} \, dt

    \displaystyle u = \sin\left(\frac{t}{3}\right)

    \displaystyle du = \frac{1}{3} \cos\left(\frac{t}{3}\right) \, dt

    \displaystyle 3 \int \frac{1}{3} \cdot \frac{\cos\left(\frac{t}{3}\right)}{\sin\left(\fra  c{t}{3}\right)} \, dt

    substitute ...

    \displaystyle 3 \int \frac{du}{u}

    integrate and back substitute.
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  5. #5
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    I've got it now, thanks!
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