# Thread: Integration by substitution help

1. ## Integration by substitution help

Use the substitution x = sin\theta to find the exact value of;

$\int^{\frac{1}{2}}_0 \frac{1}{( {1-x^{2}})^\frac{3}{2}}$

$\int \frac{1}{(1-sin^{2}\theta)^{\frac{3}{2}}}$

$\frac{dx}{d\theta} = cos\theta$

so when I put this all together I get ;

$\int \frac{1}{(cos^{2}\theta)}^{\frac{3}{2}} cos\theta d\theta$

and there does not see to be much I can do from here, cant seem to cancel the cosines,

and the 3/2 power is suppose to be at the bottom, of the fraction. Cant seem to do it on latex

2. Try re-posting this one for now without LaTeX.

3. Or with LaTeX, now. You have to manually put in the
Code:
$$ tags around the LaTeX source code, I think. 4. ## Here... unfortunately, though, you may have to wait for the latex to work. Did you forget that (x^m)^(n/p) = x^{(mn)/p}, i.e. {(cosx)^2}^(3/2) = (cosx)^{(3*2)/2} = cos^3{x}? Full solution here, but you may have to wait for the latex to work. We have [LaTeX ERROR: Compile failed] If we let (as suggested) [LaTeX ERROR: Compile failed] , then [LaTeX ERROR: Compile failed] When [LaTeX ERROR: Compile failed] , [LaTeX ERROR: Compile failed] ; similarly, when [LaTeX ERROR: Compile failed] , [LaTeX ERROR: Compile failed] . Then we have [LaTeX ERROR: Compile failed] [LaTeX ERROR: Compile failed] [LaTeX ERROR: Compile failed] 5. Originally Posted by TheCoffeeMachine Full solution here, but you may have to wait for the latex to work. Used this page to make an image, useful for now, click on image for full size: For fast results, used search-and-replace [tex] -> }$$ -> \text{

and then check the beginning and end to make sure no missing or extra \text{ or }

6. Here is a trick , substitute x = 1/t we have dx = -dt/t^2 and it becomes

int_{oo}^{2} (-dt/t^2)/( 1 - 1/t^2 )^(3/2)

= int_{2}^{oo} [dt/t^2(t^3)] /( t^2 -1 )^(3/2)

= int_{2}^{oo} (t dt)(t^2 - 1)^(-3/2)

= [ (t^2-1)^(-1/2) (-2) / 2 ]_{2}^{oo}

= - [ 1/sqrt(t^2-1) ]_{2}^{oo}

= - 0 + 1/sqrt(2^2-1) = 1/sqrt(3)

7. Originally Posted by undefined
Used this page to make an image, useful for now, click on image for full size:

For fast results, used search-and-replace

$$-> }$$ -> \text{

and then check the beginning and end to make sure no missing or extra \text{ or }
Cool, thanks! Recreated the picture, with corrections of typos that I've (no wonder) made earlier:

8. Originally Posted by TheCoffeeMachine
Cool, thanks! Recreated the picture, with corrections of typos that I've (no wonder) made earlier:
You're welcome!

How did you get the inline image to appear full size? I see that the code is

[IMG]http://www.mathhelpforum.com/math-help/attachment.php?attachmentid=17895&d=1276832168[/IMG]

but what steps did you use to produce it?

9. Originally Posted by undefined
You're welcome!

How did you get the inline image to appear full size? I see that the code is

[IMG]http://www.mathhelpforum.com/math-help/attachment.php?attachmentid=17895&d=1276832168[/IMG]

but what steps did you use to produce it?
After uploading, click on the attachment (where it says Attached Thumbnails), and then it will take you to the image url. (When you click to view the image you posted earlier in full size, clicking again will give you the URL).

EDIT: Just realised that might not have been clear.

After uploading, I clicked 'preview post' and clicked on the attachment, then it took me to here. I copied http://www.mathhelpforum.com/math-he...5&d=1276832168 from the address bar and went back to replying, clicked on the image icon in the tool bar, and pasted it on the little box that appears, and the picture appeared full.