dummy variable in Fourier transform confusion

Question

In this text, why is it using different dummy variable for the integral of coefficients $a_n$ and $b_n$? I know that choosing the dummy variable does not affect integral but over here since we are writing Fourier series of $f(x)$ choosing $t$ as dummy variable and $x$ in the series to represent the fourier series has led to expression $(15.17)$ which seems that he is considering $x$ and $t$ to be quite different?!
I know I am missing something important, but I am confused, please help me.

Answer 1

Hint. You may observe that, from $(15.14)$ to $(15.15)$ above, the author uses the classic identity $$ \cos \frac{n\pi x}{L}\cos \frac{n\pi t}{L}+\sin \frac{n\pi x}{L}\sin \frac{n\pi t}{L}=\cos \frac{n\pi}{L}(t-x) \tag1 $$ then he makes use of a Riemann sum to obtain the final form $$ f(x)=\frac 1\pi \int_0^{\infty}d\omega\int_{-\infty}^{\infty}f(t) \cos (\omega(t-x))\: dt. \tag2 $$ In $(2)$, $t$ is a dummy variable and it is of course very different from the variable $x$, the latter being considered as a parameter for the integral.

Hoping this helps.