What is $f(x')$ in a trigonometrical series?

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I read the book The Mathematics of Diffusion by John Crank. In a case, the solution is as a trigonometrical series of

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$x$ is the distance, what is the term $f(x')$?

Shouldn't we calculate the series for any given value of $x$? Then, what is $x'$?

How can the last integral be calculated?