Advanced Integration Notes
1. Basics of Integration
Integration is the reverse process of differentiation.
If d/dx[F(x)] = f(x), then the integral of f(x) dx is F(x) + C, where C is the constant of integration.
2. Standard Integration Formulas
Integral of x^n dx = x^(n+1)/(n+1) + C, for n not equal to -1
Integral of 1/x dx = ln|x| + C
Integral of e^x dx = e^x + C
Integral of a^x dx = a^x/ln(a) + C
Integral of sin x dx = -cos x + C
Integral of cos x dx = sin x + C
Integral of sec^2 x dx = tan x + C
Integral of csc^2 x dx = -cot x + C
Integral of sec x tan x dx = sec x + C
Integral of csc x cot x dx = -csc x + C
3. Methods of Integration
A. Substitution: Used for composite functions.
Example: Integral of sin(3x) dx = -1/3 cos(3x) + C
B. By Parts: Integral of u·v dx = u * integral of v dx - integral of (du/dx * integral of v dx)
LIATE rule: Logarithmic > Inverse > Algebraic > Trig > Exponential
C. Partial Fractions: For rational functions.
Example: Integral of 1/(x^2 - 1) dx = (1/2)ln|x-1| - (1/2)ln|x+1| + C
D. Trig Substitution: For sqrt(a^2 - x^2), etc.
Example: Integral of 1/sqrt(a^2 - x^2) dx = inverse sin(x/a) + C
