| Expression | Substitution | Identity Used | Result |
|---|---|---|---|
| sqrt(a^2 - x^2) | x = a sin(theta) | 1-sin^2 = cos^2 | sqrt becomes a*cos(theta) |
| sqrt(a^2 + x^2) | x = a tan(theta) | 1+tan^2 = sec^2 | sqrt becomes a*sec(theta) |
| sqrt(x^2 - a^2) | x = a sec(theta) | sec^2-1 = tan^2 | sqrt becomes a*tan(theta) |
After substitution: Express everything in terms of theta, integrate, then convert back using a right triangle.
Common pitfall: Forgetting to change dx. If x = a sin(theta), then dx = a cos(theta) d(theta).
Example: integral dx/sqrt(x^2+4). Let x = 2tan(theta). dx = 2sec^2(theta)d(theta). sqrt(x^2+4) = 2sec(theta). Integral = integral sec(theta)d(theta) = ln|sec(theta)+tan(theta)| = ln|x/2 + sqrt(x^2+4)/2| = ln|x+sqrt(x^2+4)| - ln2.