Writing a code for symbolic integration

I'm trying to integrate the following

where

How do I write a code that will integrate this symbolically? I have been trying without success.

Answer 1

[Haley Lang]

Update, I have entered the following code:

syms T_i T_inf C lambda_one t r_o J_1 alpha

f = ((T_i)-(T_inf))*((-C*(lambda_one))/(r_o))*(J_1)*(((lambda_one)*(r_o))/((r_o)))*exp((-(lambda_one)^2*(alpha*t))/((r_o)^2));

int(f,t)

and have gotten the following as an answer:

ans =

(C*J_1*r_o*exp(-(alpha*lambda_one^2*t)/r_o^2)*(T_i - T_inf))/alpha

Anyone know if this seems correct? Much appreciated.

Answer 2

[Torsten]

Setting

C1 = (T_i-T_inf)*(-C1*lambda_1/r0)*J1(lambda_1*r0/r0) 
C2 = lambda_1^2*alpha/r0^2

your function is

f(t) = C1*exp(-C2*t)

An integral is

F(t) = -C1/C2*exp(-C2*t)

Thus

integral_{0}^{t} F(t') dt' = C1/C2*(1-exp(-C2*t))

By the way: My guess is that J1 is not a constant, but the Bessel function of the first kind.

Furthermore, I guess that T_i depends on t as well such that your integration is not corrrect.

Best wishes

Torsten.