# Question:Solution of definite integration

## Question:Solution of definite integration

Maple
```restart;
Digits := 15:
L := 1:
E := 100:
nu := 0.2:
G := E/2.6:
h := 0.1:
b := 0.1:

s := -E*(diff(w(x), x, x))*sinh(sqrt(2*Pi^2*N*(1+nu)/L^2)*y)/(sqrt(2*Pi^2*N*(1+nu)/L^2)*cosh(sqrt(Pi^2*N*(1+nu)/(2*L^2))*h)):

t := G*(diff(w(x), x))*(1-cosh(sqrt(2*Pi^2*N*(1+nu)/L^2)*y)/cosh(sqrt(Pi^2*N*(1+nu)/(2*L^2))*h)):

integrand := b*(int(t^2/(2*G)+s^2/(2*E), y = -(1/2)*h .. (1/2)*h))-(1/2)*E*b*h^3*evalf(Pi^2)*N*(diff(w(x), x))^2/(12*L^2):

integrand := subs(diff(w(x), x, x) = S, diff(w(x), x) = F, w(x) = Z, integrand):

EQ := subs(S = diff(w(x), x, x), F = diff(w(x), x), Z = w(x), diff(integrand, Z))-(diff(subs(S = diff(w(x), x, x), F = diff(w(x), x), Z = w(x), diff(integrand, F)), x))+diff(subs(S = diff(w(x), x, x), F = diff(w(x), x), Z = w(x), diff(integrand, S)), x, x):

W := rhs(dsolve(EQ)):

u1 := (int((E*sinh(sqrt(2*Pi^2*N*(1+nu)/L^2)*y)/(sqrt(2*Pi^2*N*(1+nu)/L^2)*cosh(sqrt(Pi^2*N*(1+nu)/(2*L^2))*h)))^2, y = -(1/2)*h .. (1/2)*h))*(int((diff(W, x, x))^2, x = 0 .. L))/(2*E):

u2 := (int(G^2*(1-cosh(sqrt(2*Pi^2*N*(1+nu)/L^2)*y)/cosh(sqrt(Pi^2*N*(1+nu)/(2*L^2))*h))^2, y = -(1/2)*h .. (1/2)*h))*(int((diff(W, x))^2, x = 0 .. L))/(2*G):

U := simplify(u1+u2-(1/2)*E*b*h^3*evalf(Pi^2)*N*(int((diff(W, x))^2, x = 0 .. L))/(12*L^2))```

It seems that solving of the above integration is a very time consuming process.

Please propose a way to solve above integration, if it is possible.

Thanks

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