(Integral of f(x) dx) * (Integral of 1/f(x) dx) = -1
How can I find all equations f(x) that satisfy this equation? Its obvious the f(x)=e**x works, but how can I prove this? I tried differentiating but that doen't seem to be helping. Thanks for help
(Integral of f(x) dx) * (Integral of 1/f(x) dx) = -1
How can I find all equations f(x) that satisfy this equation? Its obvious the f(x)=e**x works, but how can I prove this? I tried differentiating but that doen't seem to be helping. Thanks for help
The wording of the problem is ambiguous because we don't know if the integrals are definite or undefinite integrals.
If the integrals were definite, the bounds should be specified and the problem may have a solution, or no solution, depending on the given bounds.
If the integral were undefinite, there is no solution. For example f(x)=exp(x) doesn't works :
Integral of exp(x)*dx = exp(x)+C1
Integral of (1/exp(x))dx = -exp(-x)+C2
the product = (exp(x)+C1)*(-exp(-x)+C2) is not equal to -1.
Can you clarify the wording of the problem about the bounds of the integrals ?
Also if they are indefinite integrals, the constants should be the same since they are from the same function so they would cancel out for e^x and it would work. So I'm fairly certain now the integrals are indefinite.
I don't agree.
For example, with f(x)=exp(x) :
Integral of exp(x)*dx = exp(x)+C1
Integral of (1/exp(x))dx = -exp(-x)+C2
Suppose C1=C2=C
the product = (exp(x)+C)*(-exp(-x)+C) is not equal to -1 except if C=0.
But if C=0, the integrals are no longer undefinite integrals because you have to define some particular values of the bounds so that C could be computed and equal to 0. In that case, the integrals are definite integrals with one bound =x and with the other bound = a particular value to be defined (not necessarily the same for both integrals).
If you raise differently the problem (in taking account of the preceeding remark), I could show you how to solve it.
See the attachments.
The integrals are definite integrals.
The bounds must not be the same : The only case where they are solutions is if one of the bounds of an integral is -infinity and one of the bounds of the other integral is +infinity.
On these conditions, the solutions are y(x)=c*exp(x) or y(x)=c*exp(-x) with c=constant different from 0.
It is proved that no other solution exists.
Of course, one can solve this with a differential equation. The differential equation given at end of the attachment to post #7 is very easy to solve. The solutions are explicitly written in post #7.