# Complicated function definite integration problem

I have a function with two variables (s,t):

 AT = 0.0000792456 E^((0.08636 - 0.141381 t +
0.0169775 (-0.282843 + Sqrt[0.08 + t])) (73.3877 +
Log[s])) s (-0.212432 + 0.990813 (73.3877 + Log[s])) +
4.2148 E^(-2.13198 t) s^(0.875076 - 1.06027 t +
1.07567 (-0.282843 + Sqrt[0.08 + t])) Sin[
1/2 \[Pi] (0.875076 - 1.06027 t +
1.07567 (-0.282843 + Sqrt[0.08 + t]))]


and I have to obtain the next definite integral:

Integrate[AT, {t, 0, 1}]


which will be a function of s, but Mathematica does not calculate this integral, and I obtain the input expression after evaluation:

How can I solve this problem?

• I suspect that a numerical integral is the best that you can do, but applying assumptions about the value of s (e.g. s>0 might help) – mikado Feb 26 '17 at 22:03

If you are satisfied with a ParametricFunction object, you could use:

sol = ParametricNDSolveValue[{f'[t] == AT, f[0] == 0}, f[1], {t,0,1}, s];


The output is a ParametricFunction that gives you the value of the integral for a given value of s. For example, you can plot it:

Plot[sol[s],{s,0,1}]


Compare to evaluating the definite integral numerically:

Block[{s = .3}, NIntegrate[AT, {t, 0, 1}]]
sol[.3]


0.692984

0.692984

You could do this numerically:

f[ss_] := NIntegrate[AT /. s -> ss, {t, 0, 1}]