L[f(t)] = 1/[1-exp(-sT)]*int(exp(-st)*f(t)dt,t=0..T)
By taking the Laplace transform and using the convolution theorem, obtain the solution of the integral equation
f(t) = 1+t+int[(t-u)f(u)du,u=0..t).
Please see the attached file.
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The solution shows how to use the Laplace transform definition and convolution properties to obtain the required results.