Hi,

I am trying to integrate (x^5)/7680 . exp (-x/2) dx between 0 and 1. I've had various attempts at this and this is what i have done so far....

Taken the 1/7680 outside the integration

Using integation by parts I have assigned u=x^5 and dv/dx = exp(-x/2). when i integrate exp(-x/2) i get v = -2exp(-x/2), and du/dx = 5x^4 - is this correct?

This integration results in....

-2x^5exp(-x/2) - the integral of -8x^4 . exp(-x/2)

which also has a product so i do a series of integration of parts to nock down the power each time to get the solution in terms of a x^5 + x^4 +x^3 etc....

But i mut be doing something wrong as the answer is 1.4x10^-5 and i get 0.189!

I would really appreciate some help on this integration as I've spent so long trying to figure it out and this is my last resort!

Thanks

zxcv784

sorry there is an error in my first post...

after the first integration stage i now get...

-2x^5.exp(-x/2) -integral of -10x^4 . exp(-x/2)

after continuing the integrations my final result is:

-2x^5.exp(-x/2) - 20x^4.exp(-x/2) + 160x^3.exp(-x/2) - 960x^2.exp(-x/2) + 3840x.exp(-x/2) - 7680exp(-x/2)

I have tried to substitue the boundaries into this (x=1 and x=0) but get a minus number so i must have gone wrong somewhere, any ideas?

your help on this is greatly appreciated.

zxcv784

tiny-tim
Homework Helper
Welcome to PF!

Hi zxcv784! Welcome to PF!
-2x^5.exp(-x/2) - 20x^4.exp(-x/2) + 160x^3.exp(-x/2) - 960x^2.exp(-x/2) + 3840x.exp(-x/2) - 7680exp(-x/2)

Shouldn't they all be minuses?

i don't think (because uv - -x = uv + x) so but could be wrong....

even if they were all minuses when i subsitute x=1 into it i get -7679.54 which when divided by 7680 gives -0.999 which is still way off....

any ideas?

tiny-tim
Homework Helper
… even if they were all minuses when i subsitute x=1 into it …

D'uh … what about x = 0 ?

D'uh....when x = 0, and you substiute it into x^n the result is 0.....

I would substitute s = x/2 and then your left with

$$\frac{1}{120}\int_0^{1/2} s^5\,e^{-s}\,ds$$

which equals
$$-\frac{1}{120}\left[e^{-s}\cdot P_{5}(s)\right]_0^{\frac{1}{2}}$$

Where P(s) is a polynomial resulting from the integration by parts, which, I believe, can be shown to be in general,

$$P_n(s) = s^n + n\cdot s^{n-1} + n\cdot(n-1)\cdot s^{n-2} \ldots +n!$$

tiny-tim
Homework Helper
D'uh....when x = 0, and you substiute it into x^n the result is 0.....

Even for n = 0?