# Two questions from calc II

• Oct 9th 2005, 06:27 PM
msm1593
Two questions from calc II
hey guys, thanks to those who helped me on my last problem, this time i've ufortunately got two :(

1) for the integral x
trap(10)= 12.676
trap(30)= 10.420

i have to find the actual value

the error is n^2 when using trap so 20^2= 400, 12.676-10.420 = 2.256

so is x = 2.256/400?

2) this problem i don't really have any idea what to do whatsoever, and it's a proof (which are gods way of punishing those you take math classes)

i need to integrate R(x) = t^(x-1)e^(-t)dt from zero to infinity

then prove that R(x+1) = nRn

i tried working out the anitderivate for the start, and i came out with just (t^(x+1)e^(-t)) / ln(t)

i'm not really even sure if that is correct

thanks in advance to anyone who can help me
• Oct 15th 2005, 02:14 PM
hpe
Quote:

Originally Posted by msm1593
hey guys, thanks to those who helped me on my last problem, this time i've ufortunately got two :(

1) for the integral x
trap(10)= 12.676
trap(30)= 10.420

i have to find the actual value
the error is n^2

The error is actually of the form $\frac{C}{n^2} + O(n^{-4})$, where $C$ depends on the second derivative of f. Let's assume it is exactly $\frac{C}{n^2}$, with unknown $C$, and call the unknown integral I. Then you have the equations
$12.676 = I + \frac{C}{100}$
$10.420 = I + \frac{C}{900}$
Thus your can solve for I: $I = \frac{9 \cdot 10.420 - 12.676}{8} = 10.138$ with an error of the order $n^{-4} \sim 10^{-4}$, thus correct to two or three decimal digits.
This is called "Aitken extrapolation" and in this context as "Romberg integration", so you should give proper attribution.
Quote:

2) this problem i don't really have any idea what to do whatsoever, and it's a proof (which are gods way of punishing those you take math classes)

i need to integrate R(x) = t^(x-1)e^(-t)dt from zero to infinity

then prove that R(x+1) = nRn

i tried working out the anitderivate for the start, and i came out with just (t^(x+1)e^(-t)) / ln(t)

i'm not really even sure if that is correct
No it isn't :(
Try integration by parts in the improper integral. This is known as the Gamma function,
$\Gamma(x) = \int_0^\infty t^{x-1}e^{-t}dt$
and the formula $\Gamma(x+1) = x \Gamma(x)$ is the functional equation of the Gamma function.
• Oct 26th 2005, 07:05 PM
Jameson
Quote:

Originally Posted by msm1593
2) this problem i don't really have any idea what to do whatsoever, and it's a proof (which are gods way of punishing those you take math classes)

i need to integrate R(x) = t^(x-1)e^(-t)dt from zero to infinity

then prove that R(x+1) = nRn

i tried working out the anitderivate for the start, and i came out with just (t^(x+1)e^(-t)) / ln(t)

i'm not really even sure if that is correct

thanks in advance to anyone who can help me

Substitute x+1 into the equation you were given and integrate from there. It's not that bad.