Integrating Complicated Functions: (1-a)^s

[adnan jahan]

Hi, i am stuck in integration of a complicated functions which is as,


∫₀¹(1-a)^{s}|f′′(x)|^{(1-a)^{s}}|f′′(t)|^{a^{s}}da

where∫₀¹ is integration from 0 to 1.

waiting for your kind reply.

 

[Ulagatin]

Sorry, but could you please repost this so it can be read more clearly?

That way, we may be able to help you. : ) Also, you must show at least some working.

 

[adnan jahan]

i have uploaded the pdf containing the exact form of the integral thanks

 

[Gib Z]

You integral can be put into the form

$$\int^1_0 x^s A^{x^s} B^{ (1-x)^s} dx$$ where A and B are constants. No closed form solution for that, though for some particular combinations of s, A and B you might be able to. So unless you can tell us more about f''(a) and f''(t) and s, you are out of luck.

 

[paulfr]

There is almost certainly no Antiderivative for this function.

But it can be solved easily with numerical techniques.
If accuracy is not too important, just calculate the value of the function
for 5 or 10 values of x between 0 and 1, average those values and
that will be the value of the definite integral.

The accuracy will increase as you increase the number of points used.

Note; ordinarily one would have to multiply the average of f(x) by
the interval, but in this case the interval is = 1 - 0 = 1.

 

[HallsofIvy]

There is almost certainly no Antiderivative for this function.

There certainly is an anti-derivative just because the function is continuous. What you mean is that there is no anti-derivative in terms of elementary functions.

But it can be solved easily with numerical techniques.
If accuracy is not too important, just calculate the value of the function
for 5 or 10 values of x between 0 and 1, average those values and
that will be the value of the integral.

The accuracy will increase as you increase the number of points used.

Note; ordinarily one would have to multiply the average of f(x) by
the interval, but in this case the interval is = 1 - 0 = 1.