improper integral: int[-inf, +inf.][(x^7) / (e^(-x^8))]dx

[thebenji]

how do i do the integral from negative infinity to positive infinity of:

(x^7)(e^(-x^8))

?

 

[pka]

Can you tell us what you have tried?
What have you done on this problem?
Show us something you have done!

 

[thebenji]

sure thing.

first i separated the integral into two integrals that run from -infinity to zero and zero to +infinity

then i did tabular integration--i think this is where i messed up

U DV
x^7 e^(-x^8)
-7x^6 (1/-x^8)e^(-x^8)
42x^5 (1/-x^8)^2*e^(-x^8)
-210x^4 (1/-x^8)^3*e^(-x^8)
840x^3 (1/-x^8)^4*e^(-x^8)
-2520x^2 (1/-x^8)^5*e^(-x^8)
5040x (1/-x^8)^6*e^(-x^8)
-5040 (1/-x^8)^7*e^(-x^8)
0 (1/-x^8)^8*e^(-x^8)

after simplification, i wound up with:

g(x) = (e^(-x^8))((-1/x)+(7/x^10)-(42/x^19)+(210/x^28)-(840/x^37)+(2520/x^46)-(5040/x^55)+(5040/x^64))

i evaluated as:

lim(t --> -infinity) of integral from t to 0 of "g(x)" yielded 0

and then as:

lim(t --> infinity) of integral from 0 to t of "g(x)" yielded 0

0+0=0

hey wait, nevermind. that the correct answer.

but still, i don't think i did the tabular antiderivatives correctly.

i don't have a problem with the concept behind improper integrals--more precisely, my problem is how to go about evaluating that integral. suppose it was indefinite?

 

[thebenji]

let me make the tabular more clear:

U ______________DV
x^7 _____________e^(-x^8)
-7x^6 ___________(1/-x^8)e^(-x^8)
42x^5 ___________(1/-x^8)^2*e^(-x^8)
-210x^4 _________(1/-x^8)^3*e^(-x^8)
840x^3__________ (1/-x^8)^4*e^(-x^8)
-2520x^2_________ (1/-x^8)^5*e^(-x^8)
5040x____________ (1/-x^8)^6*e^(-x^8)
-5040 ____________(1/-x^8)^7*e^(-x^8)
0 ________________(1/-x^8)^8*e^(-x^8)

 

[tkhunny]

What on Earth are you doing?

Couldn't you try a simple substitution? \(\displaystyle u = x^{8}\), for example?

Better yet, can't you just notice that you have an odd function with limits of zero in both directions? Pretty much has zero (0) written all over it.

 

[thebenji]

Thanks. The u-substitution thing was pretty obvious.