# How to integrate this equation

## Main Question or Discussion Point

T(z,t) = ∫ ( exp(-αz)* erfc(-α*sqrt((k*t)/(c*ρ))+0.5 sqrt((c*ρ*z)/(k*t)))*exp( -((t-ζ)/ζ0)^2)

integrate with respect to ζ in the limits 0 and t

Help will be greatly appreciated

HallsofIvy
Homework Helper
I think you are letting complicated lookig constants confuse you.

If I read the parentheses correctly that is
$$\int (A+ 0.5\sqrt{Be^{-2(t- ζ)/ζ_0}})dζ$$
with A and B representing those rather complicated constants in your integral.
I presume you know that $\int Adt= At+ C$. For the second integral, let $u= -2(t- ζ)/ζ_0$, so that $dζ= (ζ_0/2)du$, so the second integral becomes
$$B\frac{\zeta_0}{2}\int e^u du$$

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Mark44
Mentor
Fixed the LaTeX in the integral.
If I read the parentheses correctly that is
$$\int (A+ 0.5\sqrt{Be^{-2(t- ζ)/ζ_0}})dζ$$
with A and B representing those rather complicated constants in your integral.
I presume you know that $\int Adt= At+ C$. For the second integral, let $u= -2(t- ζ)/ζ_0$, so that $dζ= (ζ_0/2)du$, so the second integral becomes
$$B\frac{\zeta_0}{2}\int e^u du$$