How do I evaluate: http://www.wolframalpha.com/input/?i=integral+sqrt((x^2+%2B+1)^2%2F(x^2+-+1)^2)%2C+from+0+to+1%2F2 ?
Any help would be greatly appreciated!
Thanks in advance!
$\displaystyle \sqrt{\frac{(x^2 + 1)^2}{(x^2 - 1)^2}} = \frac{x^2 + 1}{x^2 - 1}$
$\displaystyle = \frac{x^2 - 1 + 2}{x^2 - 1}$
$\displaystyle = 1 + \frac{2}{x^2 - 1}$
$\displaystyle = 1 + \frac{2}{(x - 1)(x + 1)}$.
Now by using partial fractions:
$\displaystyle \frac{A}{x - 1} + \frac{B}{x + 1} = \frac{2}{(x - 1)(x + 1)}$.
LHS: $\displaystyle \frac{A}{x - 1} + \frac{B}{x + 1} = \frac{A(x + 1)}{(x - 1)(x + 1)} + \frac{B(x - 1)}{(x - 1)(x + 1)}$
$\displaystyle = \frac{A(x + 1) + B(x - 1)}{(x - 1)(x + 1)}$.
The numerators are equal, so
$\displaystyle A(x + 1) + B(x - 1) = 2$
$\displaystyle Ax + A + Bx - B = 2$
$\displaystyle (A + B)x + A - B = 0x + 2$.
So $\displaystyle A + B = 0$ and $\displaystyle A - B = 2$.
Therefore $\displaystyle A = B + 2$
$\displaystyle B + 2 + B = 0$
$\displaystyle 2B + 2 = 0$
$\displaystyle B = -1$.
$\displaystyle A = B + 2$
$\displaystyle A = -1 + 2$
$\displaystyle A = 1$.
Therefore $\displaystyle \frac{2}{(x - 1)(x + 1)} = \frac{1}{x - 1} - \frac{1}{x + 1}$.
So your integral:
$\displaystyle \int{\sqrt{\frac{(x^2 + 1)^2}{(x^2 - 1)^2}}\,dx} = \int{1 + \frac{1}{x - 1} - \frac{1}{x + 1}\,dx}$.
You should be able to go from here.
Be careful here. Since we are integrating from 0 to 1/2, we have that $\displaystyle x^2-1<0$, so we should actually get:
$\displaystyle \sqrt{\frac{(x^2 + 1)^2}{(x^2 - 1)^2}} = - \frac{x^2 + 1}{x^2 - 1}$
This doesn't invalidate the rest of the work, but we just need to switch the sign.
In other words, we should instead use:
$\displaystyle \int_0^{1/2} \sqrt{\frac{(x^2 + 1)^2}{(x^2 - 1)^2}}\,dx ~=~ -\int_0^{1/2} \left( 1 + \frac{1}{x - 1} - \frac{1}{x + 1} \right) dx$
When using partial fraction decomposition, you need the numerator to have lower degree than the denominator. In a more complex problem you may have to use polynomial long division to see the reduction, but in this case it's pretty easy to rewrite the expression to get it.
$\displaystyle \frac{x^2 +1}{x^2 - 1} = \frac{x^2 - 1 + 2}{x^2 - 1} = \frac{x^2-1}{x^2-1} + \frac{2}{x^2-1} = 1 + \frac{2}{x^2-1}$
You attempted to use partial fraction decomposition on the following fraction
$\displaystyle \frac{-(x^2+1)}{x^2-1}$
You cannot do that! Partial fraction decomposition cannot be used on this fraction because the degree of the numerator is not less than the degree of the denominator.
Notice how you got
$\displaystyle \frac{1}{x+1} - \frac{1}{x-1}$
when you did the work. But now, let's check the answer:
$\displaystyle \frac{1}{x+1} - \frac{1}{x-1} = \frac{1}{x+1} \left( \frac{x-1}{x-1} \right) - \frac{1}{x-1} \left( \frac{x+1}{x+1} \right)$
$\displaystyle = \frac{x-1}{(x+1)(x-1)} - \frac{x+1}{(x-1)(x+1)}= \frac{x-1-(x+1)}{x^2-1} = \frac{-2}{x^2-1}$
But this doesn't equal the original fraction.