Reciprocal substitution (x = 1/t): Useful for integral dx/(x^n * sqrt(ax^2+bx+c)) or integrals where x appears in denominator with high power.

Example: integral dx/(x^2 * sqrt(x^2+1)). Let x = 1/t, dx = -dt/t^2. Integral becomes integral (-dt/t^2)/((1/t^2)sqrt(1/t^2+1)) = -integral tdt/sqrt(1+t^2) = -sqrt(1+t^2) = -sqrt(1+1/x^2) + C = -sqrt(x^2+1)/x + C.

Substitution x^n = t: For integrals involving x^(n-1)dx with expressions in x^n.

Substitution x = asin^2(theta) + bcos^2(theta): For integral dx/sqrt((x-a)(b-x)) type. Rationalizes both factors.