The substitution method (also called u-substitution or change of variable) is the single most important integration technique. It reverses the chain rule of differentiation: if F'(g(x))*g'(x) is the derivative of F(g(x)), then integral F'(g(x))*g'(x) dx = F(g(x)) + C.

Recognition Patterns: The key skill is recognizing that the integrand contains a function and (a multiple of) its derivative. Common patterns include: x*e^(x^2) (derivative of x^2 is 2x), sin(x)*cos^n(x) (derivative of cos x is -sin x), sec^2(x)tan^n(x) (derivative of tan x is sec^2 x), and 1/(xln x) (derivative of ln x is 1/x).

Linear Substitution: For f(ax+b), simply divide by the coefficient a: integral f(ax+b) dx = F(ax+b)/a + C. This is so common it should be done mentally without explicit substitution.

Power Substitution: When the integrand involves sqrt(x), sqrt3, or fractional powers, substitute t = x^(1/n) where n is the LCM of all fractional exponent denominators. This rationalizes the integrand.

Trigonometric to Algebraic: For integrals like integral cos^5(x)*sin(x) dx, substitute t = cos(x): integral = -integral t^5 dt = -t^6/6 + C = -cos^6(x)/6 + C. The rule: save one factor of the trig function whose derivative you need, convert everything else.

Reciprocal Substitution: For integrals involving 1/x^n patterns, sometimes x = 1/t simplifies the expression. Useful for integral dx/(x^2*sqrt(x^2-1)) type problems.

Euler Substitutions: For sqrt(ax^2+bx+c), three types: (1) sqrt(ax^2+bx+c) = t + x*sqrt(a) when a > 0, (2) sqrt(ax^2+bx+c) = tx + sqrt(c) when c > 0, (3) sqrt(a(x-r)(x-s)) = t(x-r) when real roots exist.