Mathematics · Ch 10 — Integrals

Summary

Indefinite integral: $\int f(x)\,dx=F(x)+C$ where $F'(x)=f(x)$ ; ALWAYS verify by

differentiating the answer.

Substitution: $u=g(x)$ , $du=g'(x)dx$ turns $\int f(g(x))g'(x)\,dx$ into $\int f(u)\,du$ ; use

a trig identity first if needed (e.g. $\sin^3x=\sin x(1-\cos^2x)$ ).

Partial fractions: decompose a proper rational function by its factored denominator (distinct

linear $\to A/(x-a)$ ; repeated $\to$ one term per power; irreducible quadratic $\to$ linear

numerator), dividing first if improper.

By parts: $\int u\,dv=uv-\int v\,du$ , from the product rule; choose $u$ by ILATE

(Inverse trig, Log, Algebraic, Trig, Exponential).

Standard forms (algebraic denominators, derived via $x=a\tan\theta$ /partial fractions, extended by completing the square):

\int\frac{dx}{x^2+a^2}=\frac1a\tan^{-1}\frac xa+C, \qquad \int\frac{dx}{x^2-a^2}=\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C.

Standard forms (square roots, derived via $x=a\sin\theta$ etc.):

\int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac xa+C, \quad \int\frac{dx}{\sqrt{x^2\pm a^2}}=\ln\left|x+\sqrt{x^2\pm a^2}\right|+C,

\int\sqrt{a^2-x^2}\,dx=\frac x2\sqrt{a^2-x^2}+\frac{a^2}2\sin^{-1}\frac xa+C, \quad \int\sqrt{x^2-a^2}\,dx=\frac x2\sqrt{x^2-a^2}-\frac{a^2}2\ln\left|x+\sqrt{x^2-a^2}\right|+C.

A quadratic $ax^2+bx+c$ under a denominator or root is reduced to these via completing the

square; a linear numerator $px+q$ is split into a multiple of the derivative of the quadratic

plus a constant remainder.

Fundamental Theorem of Calculus (without proof): $A(x)=\int_a^xf(t)\,dt$ satisfies

$A'(x)=f(x)$ (First FTC); $\int_a^bf(x)\,dx=F(b)-F(a)$ for any antiderivative $F$ (Second FTC). …