Integral equation and constant rules

Question

I have an integral equation of the form:

$$f(x)=3+4\int_a^bf(t)~dt$$ How can I put the constants inside the integral to get something where I can apply the fundamental theorem of calculus?

Answer 1

Since $b-a=\int_a^b dt$, you have $k=\frac{k}{b-a}\int_a^b dt=\int_a^b \frac{k}{b-a} dt$ for any constant $k$.

Answer 2

Note that $f$ is a constant, so you need to solve $c = 3 + 4 (b-a)c$ (where $f(x) = c$), which reduces to $c (1-4(b-a)) = 3$. If $4 (b-a) \neq 1$, then the answer is $c={3 \over 1-4(b-a)}$, otherwise the is no solution.