trigonometric substitution

Some integrals come tangled up with square roots like sqrt(a^2 - x^2) or sqrt(x^2 + a^2), and no plain u-substitution will untie them. Trigonometric substitution is the move of replacing x with a trig function of a new angle so that the troublesome root collapses into something clean. The whole idea rides on the Pythagorean identities: 1 - sin^2 = cos^2, 1 + tan^2 = sec^2, and sec^2 - 1 = tan^2. Each identity is a key that fits a particular shape of root.

Concretely: for sqrt(a^2 - x^2) set x = a sin(theta), and the root becomes a cos(theta); for sqrt(a^2 + x^2) set x = a tan(theta), and the root becomes a sec(theta); for sqrt(x^2 - a^2) set x = a sec(theta), and the root becomes a tan(theta). You also replace dx by its differential (for x = a sin(theta), dx = a cos(theta) d(theta)), turn the whole integral into one in theta, integrate, then translate back to x using a right triangle that records the original substitution. A definite integral can instead just carry its limits along by converting them to angles.

This technique is the standard route to arc-length, the area of an ellipse, and many integrals in physics where a distance or potential involves sqrt(r^2 + z^2). The honest caveat is bookkeeping: you must track the sign of the root and the range of the angle (the principal branches), since sqrt(cos^2 theta) equals the absolute value of cos(theta), not cos(theta) itself. Choosing the angle interval where the cosine is non-negative is what keeps the algebra honest.