Determining $\int_a^b f(x)dx$ when $f(a)=f(b)$ How to integrate $\int_a^b f(x)dx$ when $f(a)=f(b)$
Can something like $\int_0^kf(x)dx=2\int_0^{k\over2}f(x)dx$ for $f(x)=f(k-x)$ be somehow used?

Even if $f(a)=f(b)$ there is no difference than usual way of integrating. Try to get antiderivative.
And $\int_0^k f(x)dx = 2\int _0^{k/2}f(x)dx$ can be only used if $f(x)=f(k-x)$ for all $x$. Since it is symmetric to $k/2$, $\int_0^{k/2}f(x)dx$ is equal to $\int _{k/2}^{k}f(x)dx$. So following is true. $$\int_0^k f(x)dx = \int_0^{k/2}f(x)dx +\int _{k/2}^{k}f(x)dx = 2\int _0^{k/2}f(x)dx$$
