Many definite integrals are made much easier by taking the

domain to be

complex, making up an appropriate

contour, and performing

contour integration with the aid of

complex analysis (

Cauchy's integral theorem,

residue calculus,

Jordan's lemma, etc.)

Not suprising, as complex analysis was invented in part to tackle all the gross integrals coming out of fluid mechanics, electromagnetism, and other scientific fields in the 1800s...