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

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

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

Substitution x = a$sin^2$(theta) + b$cos^2$(theta):** For integral dx/sqrt((x-a)(b-x)) type. Rationalizes both factors.