What is the error in my integration by parts?

[zxcv784]

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

 

[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]

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?

 

[zxcv784]

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]

… even if they were all minuses when i subsitute x=1 into it …

D'uh … what about x = 0 ?

 

[zxcv784]

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

 

[TheoMcCloskey]

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]

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

Even for n = 0? ​