Integration Notes (Algebra & Calculus)
1. Definition of Integration
Integration is the reverse process of differentiation. It helps to find the original function given its derivative.
- Indefinite Integral:
f(x) dx = F(x) + C, where F'(x) = f(x)
- Definite Integral:
[a to b] f(x) dx = F(b) - F(a)
2. Basic Integration Formulas
x^n dx = (x^(n+1))/(n+1) + C (n -1)
1/x dx = ln|x| + C
e^x dx = e^x + C
a^x dx = a^x / ln(a) + C
sin x dx = -cos x + C
cos x dx = sin x + C
sec^2 x dx = tan x + C
csc^2 x dx = -cot x + C
sec x tan x dx = sec x + C
csc x cot x dx = -csc x + C
3. Rules of Integration
- Constant Multiple Rule:
af(x) dx = a f(x) dx
- Sum/Difference Rule:
[f(x) g(x)] dx = f(x) dx g(x) dx
4. Methods of Integration
a. Substitution Method:
2xcos(x^2) dx Let u = x^2 cos(u) du = sin(x^2) + C
1. Definition of Integration
Integration is the reverse process of differentiation. It helps to find the original function given its derivative.
- Indefinite Integral:
f(x) dx = F(x) + C, where F'(x) = f(x)
- Definite Integral:
[a to b] f(x) dx = F(b) - F(a)
2. Basic Integration Formulas
x^n dx = (x^(n+1))/(n+1) + C (n -1)
1/x dx = ln|x| + C
e^x dx = e^x + C
a^x dx = a^x / ln(a) + C
sin x dx = -cos x + C
cos x dx = sin x + C
sec^2 x dx = tan x + C
csc^2 x dx = -cot x + C
sec x tan x dx = sec x + C
csc x cot x dx = -csc x + C
3. Rules of Integration
- Constant Multiple Rule:
af(x) dx = a f(x) dx
- Sum/Difference Rule:
[f(x) g(x)] dx = f(x) dx g(x) dx
4. Methods of Integration
a. Substitution Method:
2xcos(x^2) dx Let u = x^2 cos(u) du = sin(x^2) + C
