# Thread: Integration with Natural Logs

1. ## 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!

2. Originally Posted by Chinnie15
1) The integral of (1/(1+2x^(1/2))dx

Is this $\displaystyle\int \frac{1}{\sqrt{1+2x}} ~dx$ or $\displaystyle\int \frac{1}{1+\sqrt{2x}} ~dx$ ??

3. The second one. Sorry about that.

4. Originally Posted by Chinnie15
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}
$

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

$\displaystyle dx = \sqrt{x} \, du
$

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

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

substitute ...

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

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

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.

5. I've got it now, thanks!