Three Standard Substitutions:
| Expression | Substitution | sqrt becomes |
|---|---|---|
| sqrt(a^2-x^2) | x = a*sin(theta) | a*cos(theta) |
| sqrt(a^2+x^2) | x = a*tan(theta) | a*sec(theta) |
| sqrt(x^2-a^2) | x = a*sec(theta) | a*tan(theta) |
General quadratic under sqrt: First complete the square. ax^2+bx+c = a[(x+b/(2a))^2 + (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 dx/(x^n*sqrt(quadratic)). Transforms x-heavy denominators into t-friendly forms.
Key formula: integral (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.