Thread: 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?

i'm not really sure how to go about this



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

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 , where depends on the second derivative of f. Let's assume it is exactly , with unknown , and call the unknown integral I. Then you have the equations


Thus your can solve for I: with an error of the order , 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.

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,

and the formula is the functional equation of the Gamma function.

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.