To integrate a * exp(-t / tau) * sinc(2 * pi * t / tau) dt from t = T0 to T1, follow these steps:

1. Use the product rule of integration to split the integral into two parts:

∫ a * exp(-t / tau) * sinc(2 * pi * t / tau) dt = a * ∫ exp(-t / tau) * sinc(2 * pi * t / tau) dt

2. Use integration by parts to integrate the first part of the split integral. Let u = exp(-t/tau) and dv = sinc(2pit/tau) dt. Then, du/dt = -1/tau * exp(-t/tau) and v = (-tau/2pi) * cos(2pi*t/tau) / (t/tau).

3. Apply the integration by parts formula:

∫ u dv = u * v - ∫ v du

Substituting in the values of u, v, du/dt, and dv/dt, we get:

a * ∫ exp(-t / tau) * sinc(2 * pi * t / tau) dt = (-a/2) * (tau/ pi) * [cos(2piT1/tau)exp(-T1/tau) - cos(2piT0/tau)exp(-T0/tau)] - a * ∫ (-1/tau) * cos(2pit/tau) * exp(-t/tau) dt

4. Simplify the remaining integral by using integration by parts again, letting u = cos(2pit/tau) and dv = exp(-t/tau) dt. Then, du/dt = (-2pi/tau) * sin(2pi*t/tau) and v = -tau * exp(-t/tau).

5. Applying the integration by parts formula again, we get:

a * ∫ exp(-t / tau) * sinc(2 * pi * t / tau) dt = (-a/2) * (tau/ pi) * [cos(2piT1/tau)exp(-T1/tau) - cos(2piT0/tau)exp(-T0/tau)] + a * [(2pi/tau) * sin(2piT1/tau) * exp(-T1/tau) - (2pi/tau) * sin(2piT0/tau) * exp(-T0/tau)] - a * ∫ (2pi/tau)^2 * sin(2pi*t/tau) * exp(-t/tau) dt

6. Evaluate the remaining integral by using integration by parts one last time, letting u = sin(2pit/tau) and dv = exp(-t/tau) dt. Then, du/dt = (2pi/tau) * cos(2pi*t/tau) and v = -tau * exp(-t/tau).

7. Applying the integration by parts formula again, we get the final result:

∫ a * exp(-t / tau) * sinc(2 * pi * t / tau) dt = (-a/2) * (tau/ pi) * [cos(2piT1/tau)exp(-T1/tau) - cos(2piT0/tau)exp(-T0/tau)] + a * [(2pi/tau) * sin(2piT1/tau) * exp(-T1/tau) - (2pi/tau) * sin(2piT0/tau) * exp(-T0/tau)]
spam fails_task lang_mismatch pii not_appropriate hate_speech sexual_content quality toxicity humor helpfulness creativity violence