Integration by Substitution

• Oct 30th 2008, 02:59 PM
Apex
Integration by Substitution
Hi

Integral, I = -4 * dx/(3+2x-x^2)^0.5
There is an integral sign between 4 and dx.

I = -4 * dx/((4-(x-1)^2)^0.5 <--- completing the square

Let x-1 = 2sin(#), where # is theta

The result is 4arcsin((x-1)/2) + c, where c is an arbitrary constant

Thanks
• Oct 30th 2008, 03:56 PM
BoboStrategy
$
-4 * \int{\frac{dx}{\sqrt{((4-(x-1)^2))}}} = -4 * \int{\frac{dx}{\sqrt{((2^2-(x-1)^2))}}}
$

Then, since
$
\int{\frac{du}{\sqrt{a^2-u^2}}} = arcsin \frac{u}{a} + C,
$

just let $u = x - 1$ and then we have

$
-4 * \int{\frac{dx}{\sqrt{((2^2-(x-1)^2))}}} = -4 * arcsin \frac{x-1}{2} + C
$
• Oct 30th 2008, 04:14 PM
Apex
Thanks!
• Oct 30th 2008, 04:47 PM
Apex
The method in the notes is "let (x-1) = 2sin(theta)" for the above.

How does this work?
Can you show this using Pythagoras theorem?
Why is it arcsin rather than sin?

Thanks again
• Oct 30th 2008, 05:01 PM
BoboStrategy
Re: integration
I don't see how letting

$
(x-1) = 2 * sin(\theta)
$

is useful in this context.