Three Standard Substitutions:

General quadratic under sqrt: First complete the square. $ax^{2+bx+c}$ = a[(x+$\frac{b}{2a}$)^2 + $\frac{4ac-b^2}{(4a^2)}$]. Then apply the appropriate substitution.

Linear irrationals sqrt(ax+b): Simply substitute t = sqrt(ax+b) or $t^2$ = ax+b. Express x and dx in terms of t.

Euler substitutions (advanced): For sqrt($ax^{2+bx+c}$):

• If a > 0: let sqrt($ax^{2+bx+c}$) = t + sqrt(a)*x
• If c > 0: let sqrt($ax^{2+bx+c}$) = tx + sqrt(c)
• If real roots exist: let sqrt(a(x-alpha)(x-beta)) = t(x-alpha) These rationalize the integral but produce complex expressions.

Reciprocal substitution x = 1/t: Useful for integrals like $\frac{dx}{x^n*sqrt(quadratic}$). Transforms x-heavy denominators into t-friendly forms.

Key formula: integral $\frac{px+q}{sqrt}$($ax^{2+bx+c}$) dx. Write px+q = A*(2ax+b) + B. The first part integrates to A2sqrt(...), the second part is a standard form.