how to integrate this?

Question

how do u find the integral of t^(-1/2) * e^(-st) dt where s is any constant.

if u do integration by parts, u get stuck again so i didnt get anywhere with that. i know what the answer should be but i need to use the definition toget there.

Answer 1

If you are doing Laplace transforms (classic place for t and s) you get square root of Pi/s. Maple output:

> with(inttrans);

[addtable, fourier, fouriercos, fouriersin, hankel, hilbert,

invfourier, invhilbert, invlaplace, laplace, mellin]

> laplace(t^(-1/2),t,s);

1/2
Pi
-----
1/2
s

Answer 2

The indefinite integral of that function is only expressible using non-elementary functions, such as the error function. However, given the form, I suspect (as the other posters have mentioned) that you are doing a Laplace transform of 1/√t, in which case we only need the definite integral:

[0, ∞]∫t^(-1/2)e^(-st) dt

In this case, consider the substitution:

t=u²/s, dt=2u/s du

Which gives:

[0, ∞]∫1/√t e^(-u²) 2u/s du (note that this particular substitution does not change the limits of integration)

But 1/√t = √s/u, so this is:

[0, ∞]∫2/√s e^(-u²) du
2/√s [0, ∞]∫e^(-u²) du

But clearly, the function e^(-u²) is symmetric about the y-axis, so [0, ∞]∫e^(-u²) du = [-∞, 0]∫e^(-u²) du = 1/2 [-∞, ∞]∫e^(-u²) du. Thus you have:

1/√s [-∞, ∞]∫e^(-u²) du

And the latter quantity is of course the gaussian integral, with value √π. Thus you have:

[0, ∞]∫t^(-1/2)e^(-st) dt = √(π/s)

ETA: for a proof of the value of the gaussian integral, see the wikipedia article http://en.wikipedia.org/wiki/Gaussian_integral

Answer 3

Are you doing a laplace transform?

When I tried just to get the general answer I get an infinite series that is not any recognizable function to me.

However if s=1 and and improper integral is taken over the positive x
domain the function is the gamma function of 1/2 which is the square root of pi = 1.77245.... If this rings a bell then look into the gamma function.

Answer 4

integral of udv = uv - int(v)du

u = t^-1/2
du = -1/2t^-3/2

v=e^(-st)
dv = 1/-s(e^-st)

write in the that form as above with those values evualate and your done. I hate doing math on computer

Answer 5

well i think you hav to use integration by parts