Could you possibly use the exact value of the indefinite integral to help trying to find your solution

In[1]:= lmean = 8814/10^4;
lc = 5613/10^4;
Af = 12/10;
u = 1/10;
a = 10813/10^4;
b = 12/10;
p = 1;
q = 25294/10^4;
lct[t_] := lc*(1 - A*Tan[t])/(Exp[u*t]);
f[l_] := a*b*l^(b - 1)*Exp[-a*l^b];
g[t_] := ((Sin[t])^(2*p - 1)*(Cos[t])^(2*q - 1))/(Integrate[(Sin[w])^(2*p - 1)*(Cos[w])^(2*q-1), {w,0,Pi/2}]);
FullSimplify[Integrate[f[l]*g[t]*(l/lmean)*(l/(2*lc))*Exp[u*t], l], Assumptions -> 0 <= t < 2 Pi && 0 <= l]

Out[12]= -(1/2407281094647)126470000 E^(-((10813 l^(6/5))/10000) + t/10) l^(4/5) Cos[t]^(10147/2500)
    (150000 + 97317 l^(6/5) + 100000 E^((10813 l^(6/5))/10000)ExpIntegralE[1/3, (10813 l^(6/5))/10000]) Sin[t]

In[13]:= FullSimplify[Integrate[f[l]*g[t]*(l/lmean)*(l/(2*lc))*Exp[u*t], t, l],
   Assumptions -> 0 <= t < 2 Pi && 0 <= l]

Out[13]= -(1/(Cos[t]^(2353/2500)))(2470117187500/892485605516233971387177524702941121272167 -
    (1788364843750 I)/68652738885864151645167501900226240097859) E^(-((10813 l^(6/5))/10000) +
    (1/10 - I) t) (1 + E^(2 I t)) l^(4/5) (150000 + 97317 l^(6/5) + 100000 E^((10813 l^(6/5))/10000)
    ExpIntegralE[1/3, (10813 l^(6/5))/10000]) (1243209104158084056000 E^(I t)
    Hypergeometric2F1[2647/5000 - I/20, 1, 7353/5000 - I/20, -E^(2 I t)] - (250 + 2353 I)
    (51882557652852180070 Cos[t] + 26149933547065465035 Cos[3 t] + 5233573221864243007 Cos[5 t] - 
    250 (19936133122065382 Sin[t] + 7069109 (485993763 Sin[3 t] + 58539109 Sin[5 t]))))

If I include A=1 then I get a ConditionalExpression depending on (1 - Tan[t])^(2/5)>0 for the exact definite integral for l. For a few other values of A I get a string of "PolynomialGCD::lrgexp: Exponent is out of bounds for function PolynomialGCD." and no result.

Hi,

one of the problems in both notebooks is that you have not declared A in the equation for lct[t]. The numerical integration will not work like that. Mathematica says that the integral does not converge sufficiently well, and gives three suspected reasons for it:

Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

The integrand appears to be very nasty indeed.

If I change your 3.1416 to Pi

ClearAll["Global`*"]



lmean = 0.8814;
lc = 0.5613;
Af = 1.2;
u = 0.1;
a = 1.0813;
b = 1.2;
p = 1;
q = 2.5294;

lct[t_] := lc*(1 - Af*Tan[t])/(Exp[u*t])

f[l_] := a*b*l^(b - 1)*Exp[-a*l^b]

g[t_] := ((Sin[t])^(2*p - 1)*(Cos[t])^(2*q - 1))/
  Chop[(Integrate[(Sin[w])^(2*p - 1)*(Cos[w])^(2*q - 1), {w, 0, 
      Pi/2}])]

NIntegrate[
 f[l]*g[t]*(l/lmean)*(1 - 
    Af*Tan[t])*(1 - (lc*(1 - Af*Tan[t])/(2*l*Exp[u*t]))), {t, 0, 
  Pi/2}, {l, 0, lct[t]}]

I get

During evaluation of In[184]:= NIntegrate::zeroregion: Integration region {{1.,0.99999999999999415148564447489073961419065634827050978249802712372},{0.25,0.5}} cannot be further subdivided at the specified working precision. NIntegrate assumes zero integral there and on any further indivisible regions. >>

During evaluation of In[184]:= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. >>

During evaluation of In[184]:= NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 18 recursive bisections in t near {t,l} = {0.99999999998256007485943732904080818950806429498343042006069936178,0.499962}. NIntegrate obtained -6.7530887552778382916336050578428251258689827762888800560827789037310^893804534628964+5.4691121715425751471977471775999760599853203693952910350123540463610^893804534628964 I and -6.7530887552778382916336050578428251258689827762888800560827789037310^893804534628964+5.4691121715425751471977471775999760599853203693952910350123540463610^893804534628964 I for the integral and error estimates. >>

Out[184]= -6.75308875527783810^893804534628964 + 5.46911217154257510^893804534628964 I

In fact, if you set the Precision to infinity

lmean = SetPrecision[0.8814, Infinity];
lc = SetPrecision[0.5613, Infinity];
Af = 6/5;
u = 1/10;
a = SetPrecision[1.0813, Infinity];
b = SetPrecision[1.2, Infinity];
p = 1;
q = SetPrecision[2.5294, Infinity];

lct[t_] := lc*(1 - Af*Tan[t])/(Exp[u*t])

f[l_] := a*b*l^(b - 1)*Exp[-a*l^b]

g[t_] := ((Sin[t])^(2*p - 1)*(Cos[t])^(2*q - 
       1))/(Integrate[(Sin[w])^(2*p - 1)*(Cos[w])^(2*q - 1), {w, 0, 
     Pi/2}])

NIntegrate[f[l]*g[t]*(l/lmean)*(l/(2*lc))*Exp[u*t], {t, 0, Pi/2}, {l, 0, lct[t]}]

You get - at least in MMA10- an error message. If you click on "show all" you get a rather detailed output that shows what is going on.

Cheers, Marco

Eqn 1 is separated into two sections and notebooks are attached.

Attachments: