term-by-term integration of a Fourier series
Integration is the gentle, forgiving cousin of differentiation. Where differentiating a Fourier series term by term can blow up, integrating it term by term almost always works — you simply find the antiderivative of each sine and cosine, add them up, and you obtain the integral of the original function. This makes term-by-term integration a reliable everyday tool, the safe direction in which to push a Fourier series.
Why is it so well-behaved? Integrating cos(n pi x / L) gives sin(n pi x / L) divided by (n pi / L), so each coefficient picks up a factor of 1/n — the high harmonics are damped, not amplified, and convergence can only improve. The precise statement: if f is merely integrable over the period (it may even have jumps), then you can integrate its Fourier series term by term, and the result converges to the integral of f. There is just one bookkeeping subtlety: the constant term a_0/2 integrates to a straight line (a_0/2) x, which is not periodic, so the integrated series equals the integral of f plus that linear ramp — you fold the ramp in explicitly rather than expecting a pure Fourier series back.
This is genuinely useful, not just reassuring. You can integrate a known Fourier series to discover a new one for free, you can justify the Fourier-series steps in solving an integral equation, and you can use it to derive coefficient-sum identities. It is the operation behind getting smoother and smoother representations: integrate once and a sawtooth-like 1/n series becomes a parabola-like 1/n^2 series that converges uniformly and looks visibly nicer.
Integrating the sawtooth series x = 2[ sin x - sin 2x/2 + ... ] from 0 to x gives x^2/2 = a constant - 2[ cos x - cos 2x/4 + cos 3x/9 - ... ]: the coefficients improve from 1/n to 1/n^2, and a constant of integration appears to absorb the average.
Integration upgrades 1/n coefficients to 1/n^2 — convergence gets better, exactly the opposite of differentiation.
Do not forget the non-periodic ramp: if the mean a_0/2 of f is nonzero, the term-by-term integral contains a linear (a_0/2) x term, so the result is the integral of f, not a clean periodic Fourier series, unless that mean is zero.