Using residue theorem to compute integrals over real line intervals is an example of solving a problem by considering it in a more general setting: usually, the integrand is complexified and a closed contour is built by attaching a semicircle to the interval [-a,a].Then the integral over the contour is computed using the residue theorem and the original integral is obtained as the limit of contour integrals. This works e.g. for the function $$\int_{-\infty}^{+\infty}\frac{e^{itx}}{x^2+1}dx$$ In some cases the contour gets more complicated, to avoid branch points, as when computing $$\int_0^{\infty}\frac{dx}{x^a+1}, \quad a>1$$ Sometimes the integral over an interval is replaced by an integral over the unit circle, e.g., for $$\int_0^{\pi}\frac{d\theta}{a+\cos \theta}, \quad a>1$$ (here one also uses the equality $\cos z = (1/2)(z+1/z))$. Ahlfors's text in complex analysis explains this method in more detail. (Some other texts seem to have just a haphazard collection of examples following the statement and proof of residue theorem.)

Using residue theorem to compute integrals over real line intervals is an example of solving a problem by considering it in a more general setting: usually, the integrand is complexified and a closed contour is built by attaching a semicircle to the interval [-a,a].Then the integral over the contour is computed using the residue theorem and the original integral is obtained as the limit of contour integrals. This works e.g. for the function $$\int_{-\infty}^{+\infty}\frac{e^{itx}}{x^2+1}dx$$ In some cases the contour gets more complicated, to avoid branch points, as when computing $$\int_0^{\infty}\frac{dx}{x^a+1}, \quad a>1$$ Sometimes the integral over an interval is replaced by an integral over the unit circle, e.g., for $$\int_0^{\pi}\frac{d\theta}{a+\cos \theta}, \quad a>1$$ (here one also uses the equality $\cos z = (1/2)(z+1/z))$. Ahlfors's text in complex analysis explains this method in more detail. (Some other texts seem to have just a haphazard collection of examples following the statement and proof of residue theorem.)

This is not so much of proving a stronger result first, but rather making a problem tractable at all by using a more general approach (replacing real functions with complex ones and computing residues instead of actually integrating over the contours).

Using residue theorem to compute integrals over real line intervals is an example of solving a problem by considering it in a more general setting: usually, the integrand is complexified and a closed contour is built by attaching a semicircle to the interval [-a,a].Then the integral over the contour is computed using the residue theorem and the original integral is obtained as the limit of contour integrals. This works e.g. for the function $$\int_{-\infty}^{+\infty}\frac{e^{itx}}{x^2+1}dx$$ In some cases the contour gets more complicated, to avoid branch points, as when computing $$\int_0^{\infty}\frac{dx}{x^a+1}, \quad a>1$$ Sometimes the integral over an interval is replaced by an integral over the unit circle, e.g., for $$\int_0^{\pi}\frac{d\theta}{a+\cos \theta}, \quad a>1$$ (here one also uses the equality $\cos z = (1/2)(z+1/z))$.

Using residue theorem to compute integrals over real line intervals is an example of solving a problem by considering it in a more general setting: usually, the integrand is complexified and a closed contour is built by attaching a semicircle to the interval [-a,a].Then the integral over the contour is computed using the residue theorem and the original integral is obtained as the limit of contour integrals. This works e.g. for the function $$\int_{-\infty}^{+\infty}\frac{e^{itx}}{x^2+1}dx$$ In some cases the contour gets more complicated, to avoid branch points, as when computing $$\int_0^{\infty}\frac{dx}{x^a+1}, \quad a>1$$ Sometimes the integral over an interval is replaced by an integral over the unit circle, e.g., for $$\int_0^{\pi}\frac{d\theta}{a+\cos \theta}, \quad a>1$$ (here one also uses the equality $\cos z = (1/2)(z+1/z))$. Ahlfors's text in complex analysis explains this method in more detail. (Some other texts seem to have just a haphazard collection of examples following the statement and proof of residue theorem.)