# How can I solve this Integral?

I have a two variables function. I have to numerically or parametric integrate this function with respect to one of the variables (\tau) and assume the other variable (t) is constant. Note, however, that the upper limit of the integral is t.

λ0 = 2*Pi*10*10^6; t0 = (Sqrt[2]*Pi)/λ0; T = 250/10^9;
σ = T/8;
g0[t_, τ_] = (λ0/Sqrt[2])*Sech[((t - τ) - 0.5*T)/σ];
s[t_, τ_] = 1 + Tanh[((t - τ) - 0.5*T)/σ];
g1[t, τ] = g0[t, τ]*Sin[(Pi/4)*s[t, τ]];

NIntegrate[g1[t, τ], {τ, 0, t}] or
Integrate[g1[t, τ], {τ, 0, t}]

• t = 2; \[Lambda]0 = 2*Pi*10*10^6; t0 = (Sqrt[2]*Pi)/\[Lambda]0; T = 250/10^9; \[Sigma] = T/8; g0[t_, \[Tau]_] := (\[Lambda]0/Sqrt[2])* Sech[((t - \[Tau]) - 0.5*T)/\[Sigma]] g0[2,0.5] performs "General::munfl: Sech[4.8*10^7] is too small to represent as a normalized machine number; precision may be lost." and0,. Oct 2 '20 at 16:16
• Welcome to Mathematica Stack Exchange! I fear that what you wish cannot be done for symbolic t. You need to choose a specific value for t. Oct 2 '20 at 16:17

Integrate[(λ0/Sqrt[2])*Sech[((t - τ) - 1/2*T)/σ] (1 +
Tanh[((t - τ) - 1/2*T)/σ]), τ]


gives

(*1/Sqrt[2]λ0 σ (ArcTan[Sinh[(-2 t+T+2 τ)/(2 σ)]]+Sech[(-2 t+T+2 τ)/(2 σ)]) *)


then enter the limits.

Numerical evaluation may be difficult for your parameter values

• Thank you Andreas. But how did Mathematica solve the analysis? I did everything I could not.HOW?! Oct 3 '20 at 10:39
• replace 0.5 with 1/2 and try Integrate[g0[t, Tau]*s[t, Tau], Tau] Oct 3 '20 at 11:19
• with Plot[{((5*Pi)/(8*Sqrt[2]))*(-ArcTan[Sinh[4 - 32000000*t]] + Gudermannian[4] + Sech[4] - Sech[4 - 32000000*t]), (5* Pi*(Pi/2 + Gudermannian[4] + Sech[4]))/(8*Sqrt[2])}, {t, 0, 4/10^7}]you can plot the result using your parameter values Dec 1 '20 at 22:04