Integrals
Integral
The definite integral of a function f(x) over the interval [a, b] is:
\int_a^b f(x) \, dx
This represents the signed area under the curve of f(x) from a to b.
The indefinite integral (or antiderivative) is:
\int f(x) \, dx = F(x) + C
where F'(x) = f(x) and C is an arbitrary constant of integration.
Integration Techniques
Table of Integration Techniques.
| Rule | Function f(x) | Integral \int f(x)\,dx | Notes |
|---|---|---|---|
| Constant | c | cx + C | c \in \mathbb{R} |
| Power | x^n (with n \neq -1) | \frac{x^{n+1}}{n+1} + C | Power rule for n \neq -1 |
| Sum/Difference | g(x) \pm h(x) | \int g(x)\,dx \pm \int h(x)\,dx | Integral distributes over sums and differences |
| Exponential (base e) | e^x | e^x + C | Natural exponential |
| Exponential (base a) | a^x | \frac{a^x}{\ln a} + C | a > 0,\ a \neq 1 |
| Logarithmic | \frac{1}{x} | \ln|x| + C | Valid for x \neq 0 |
| Chain Rule (u-sub) | f(g(x))g'(x) | \int f(u)\,du, where u = g(x) | Substitution method |
| Integration by Parts | u(x)v'(x) | u(x)v(x) - \int u'(x)v(x)\,dx | \int u\,dv = uv - \int v\,du |