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: xe^($x^2$) (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 $\frac{1}{x*ln x}$ $\frac{derivative of ln x is 1}{x}$.

Linear Substitution: For f(ax+b), simply divide by the coefficient a: integral f(ax+b) dx = F$\frac{ax+b}{a}$ + 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^$\frac{1}{n}$ 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$$\frac{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 $\frac{dx}{x^2*sqrt(x^2-1}$) 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.