Mathematics · Ch 7 — Integrals

Summary

Integration as anti-derivative: If $\frac{d}{dx}F(x) = f(x)$ , then $\int f(x)\,dx = F(x) + C$ , where $C$ is the constant of integration.
Basic formulas: Memorise standard integrals like $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ ( $n \neq -1$ ), $\int e^x\,dx = e^x + C$ , $\int \frac{1}{x}\,dx = \ln|x| + C$ , and trigonometric integrals ( $\int \sin x\,dx = -\cos x + C$ , etc.).
Methods of integration:
- Substitution: Let $u = g(x)$ , then $\int f(g(x))g'(x)\,dx = \int f(u)\,du$ .
- Integration by parts: $\int u\,dv = uv - \int v\,du$ , choosing $u$ using the ILATE rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential).
- Partial fractions: For rational functions $\frac{P(x)}{Q(x)}$ , decompose into simpler fractions before integrating.
Definite integrals: $\int_a^b f(x)\,dx = F(b) - F(a)$ , where $F$ is an anti-derivative of $f$ (Newton-Leibniz formula).
Properties of definite integrals:
- $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$
- $\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx$
- $\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$
- $\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$ …