How can I reverse-engineer the process of designing a definite integral with no elementary antiderivative yet a fixed solution?
To make this concrete, assume I want to design a function $f$ such that $$I = \int_a^b f(x) \, dx=\arctan(\phi)$$ where $\phi$ denotes the golden ratio. How do I proceed in designing such a function? I will place no restrictions on the limits of integrations (except for those below), it may well be an improper integral.
Edit:I do not want to design a trivial result such as simply multiplying both sides of a known identity by the solution. Also, the bounds of integrations and the integrand mustn‘t involve $\phi$ in this case.
