# Using NIntegrate with a combined separable function

I have the function

w[x, t] = p1[x]*u1[t] + p2[x]*u2[t] + p3[x]*u3[t] + ....


where p1[x], p2[x] and p3[x] are known.

My problem is in using NIntegrate with

NIntegrate[w[x, t]*p1[x], {x, 0, 1}]


This does not work because there is the function u[t]. Is there any way to separate u[t] from each term before integration and then multiply them after preforming the integral? Or can I do NIntegrate with u[t] inside?

This can be done as follows. Here is your expression to be integrated:

w[x, t] = p1[x]*u1[t] + p2[x]*u2[t] + p3[x]*u3[t];


Assume that you need to integrate over x from 0 to 1. Then this makes the job:

w[x, t] /. u_[x]*v_[t] :> v[t]*NIntegrate[u[x], {x, 0, 1}]


yielding the following answer:

(*  NIntegrate[p1[x], {x, 0, 1}] u1[t] +
NIntegrate[p2[x], {x, 0, 1}] u2[t] +
NIntegrate[p3[x], {x, 0, 1}] u3[t]   *)


If you do not fix the functions pi[x] there will be also warnings, they should disappear as soon as Mma "knows" these functions.

Later edit: The author commented that it does not work in the following case:

p1[x_] := -Cos[4.7300407 x] + Cosh[4.7300407 x] +
0.9825022 Sin[4.7300407 x] - 0.9825022 Sinh[4.7300407 x];
p2[x_] := -Cos[7.8532046 x] + Cosh[7.8532046 x] -
1.0007773 (-Sin[7.8532046 x] + Sinh[7.8532046 x]);


Indeed, after the substitution of the rule the result is wrong, and I do not see, why.

One can go another way around, much more simple which is possible, if one has fixed the functions p1[x] and p2[x]. Namely,

 w[x_, t_] := p1[x]*u1[t] + p2[x]*u2[t];
Integrate[w[x, t], {x, 0, 1}]

(*  0.830862 u1[t] + 1.94833*10^-6 u2[t]  *)


Second edit to address your last question

OK, if you strongly need NIntegrate to accelerate the calculation, the following can be done. Let me first redefine the functions to include such that your mentioned in your question:

p1[x_] := -Cos[4.7300407 x] + Cosh[4.7300407 x] +
0.9825022 Sin[4.7300407 x] - 0.9825022 Sinh[4.7300407 x];
p2[x_] := -Cos[7.8532046 x] + Cosh[7.8532046 x] -
1.0007773 (-Sin[7.8532046 x] + Sinh[7.8532046 x]);
p3[x_] := p1[x]^3*p2[x]^4;

w[x_, t_] := p1[x]*u1[t] + p2[x]*u2[t] + p3[x]*u3[t];


You are right that using simply Integrate[w[x, t], {x, 0, 1}] takes too much time, which is unacceptable. Using NIntegrate[w[x, t], {x, 0, 1}] instead is impossible. Let us try this:

Plus @@ Table[NIntegrate[Drop[w[x, t][[i]], -1], {x, 0, 1}]*
Last[w[x, t][[i]]], {i, 1, Length[w[x, t]]}] // AbsoluteTiming

(*  {4.98792, 0.830862 u1[t] + 1.94833*10^-6 u2[t] + 2.15982 u3[t]}  *)


which takes 5 seconds. I hope that's better.

Have fun!

• in this step: w[x, t] /. u_[x]*v_[t] :> v[t]*NIntegrate[u[x], {x, 0, 1}] why did you change the pi[x] to u[x] and u[t] to v[t] ? cant i use the same names? also can you give what does the code means ? Nov 7, 2016 at 14:01
• @Love Eva This belongs to basic structures of the Wolfram Language. Have a look here wolfram.com/language/fast-introduction-for-programmers/patterns as well as Menu/Help/WolframDocumentation/guide/Patterns There you will find some more information on it. I can also recommend to read the corresponding section of the book of Leonid Shifrin which you can get here mathprogramming-intro.org for free. Nov 7, 2016 at 14:16
• is there any way to define this as rule? because w[x,t] is repeated many times and each time i should do NIntegrate. Nov 7, 2016 at 14:48
• @Love Eva It is a rule. But you may, of course write: rule=u_[x]*v_[t] :> v[t]*NIntegrate[u[x], {x, 0, 1}]. and use it everywhere. Nov 7, 2016 at 15:06
• for some reason its not working: (rule) rule2 = u_[x]*v_[t] :> v[t]*NIntegrate[u[x], {x, 0, 1}]; (functions) p1[x] = -Cos[4.7300407 x] + Cosh[4.7300407` x] + 0.9825022 Sin[4.7300407 x] - 0.9825022 Sinh[4.7300407 x]; p2[x] = -Cos[7.8532046 x] + Cosh[7.8532046 x] - 1.0007773 (-Sin[7.8532046 x] + Sinh[7.8532046 x]); w[x, t] = p1[x]*u1[t] + p2[x]*u2[t]; w[x, t] /. rule2 (-Cos[4.73004 x] + Cosh[4.73004 x] + 0.982502 Sin[4.73004 x] - 0.982502 Sinh[4.73004 x]) u1[ t] + (-Cos[7.8532 x] + Cosh[7.8532 x] - 1.00078 (-Sin[7.8532 x] + Sinh[7.8532 x])) u2[t] Nov 7, 2016 at 16:56