Integrals — Indefinite and Definite

A function $F$ is a primitive (antiderivative) of $f$ on an interval if $F'(x)=f(x)$ for every $x$. The indefinite integral $\int f(x),dx = F(x)+C$ represents the family of all primitives ($C\in\mathbb{R}$ arbitrary constant). Two primitives of the same function differ by a constant.

Fundamental indefinite integrals (memorise):

• $\int x^n,dx = \dfrac{x^{n+1}}{n+1}+C$ for $n\neq-1$
• $\int \dfrac{1}{x},dx = \ln|x|+C$ (absolute value extends the domain to $x\neq0$)
• $\int e^x,dx = e^x+C$; $\int a^x,dx = a^x/\ln a + C$
• $\int \sin x,dx = -\cos x+C$; $\int \cos x,dx = \sin x+C$
• $\int \dfrac{1}{1+x^2},dx = \arctan x+C$; $\int \dfrac{1}{\sqrt{1-x^2}},dx = \arcsin x+C$
• $\int \tan x,dx = -\ln|\cos x|+C$; $\int \cot x,dx = \ln|\sin x|+C$

Linearity: $\int(\alpha f+\beta g),dx = \alpha\int f,dx + \beta\int g,dx$.

Integration by parts: derived from the product rule: $(uv)'=u'v+uv'$. Integrating: $\int u,dv = uv - \int v,du$. Choose $u$ and $dv$ so that $\int v,du$ is simpler than the original integral. The LIATE rule suggests the order of preference for $u$: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.

Integration by substitution (change of variable): set $t=g(x)$, so $dt=g'(x)dx$. The integral becomes $\int f(g(x))g'(x),dx = \int f(t),dt$. After integrating in $t$, substitute back $t=g(x)$. Useful when the integrand contains a composite function and its derivative.

Partial fraction decomposition: for rational functions $P(x)/Q(x)$ with $\deg P < \deg Q$, factor the denominator and decompose into simpler fractions:

$$\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}, \qquad \frac{1}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}.$$

Each term can then be integrated using fundamental formulas. For irreducible quadratic denominators ($x^2+px+q$ with negative discriminant), complete the square and use arctangent.

The definite integral $\int_a^b f(x),dx$ is defined as the limit of Riemann sums:

$$\int_a^b f(x),dx = \lim_{n\to\infty}\sum_{k=1}^n f(x_k^*)\Delta x, \qquad \Delta x = \frac{b-a}{n}.$$

Geometrically, it measures the signed area between the graph of $f$ and the $x$-axis on $[a,b]$ (positive where $f>0$, negative where $f<0$).

Fundamental Theorem of Calculus (Part 1 — differentiation of the integral): if $f$ is continuous on $[a,b]$, then $F(x)=\int_a^x f(t),dt$ is differentiable and $F'(x)=f(x)$. Differentiation and integration are inverse operations.

Fundamental Theorem (Part 2 — Newton–Leibniz formula): if $F$ is any primitive of $f$, then $\int_a^b f(x),dx = F(b)-F(a)$. This reduces computing definite integrals to finding primitives.

Properties: linearity; additivity over intervals ($\int_a^c = \int_a^b + \int_b^c$); if $f\geq0$ then $\int_a^b f \geq 0$; integral mean value theorem: $\exists,\xi\in(a,b)$ such that $\int_a^b f,dx = f(\xi)(b-a)$.

An integral is improper when the interval of integration is unbounded, or when $f$ has a singularity (becomes infinite) at a point of the interval.

Unbounded interval: $\int_a^{+\infty} f(x),dx = \lim_{R\to+\infty}\int_a^R f(x),dx$. The integral converges if this limit is finite.

Singularity at an endpoint: if $f$ has a vertical asymptote at $b$, then $\int_a^b f(x),dx = \lim_{\varepsilon\to0^+}\int_a^{b-\varepsilon}f(x),dx$.

Key examples:

• $\int_1^{+\infty} x^{-\alpha},dx = \dfrac{1}{\alpha-1}$ for $\alpha>1$; diverges for $\alpha\leq1$.
• $\int_0^1 x^{-\alpha},dx = \dfrac{1}{1-\alpha}$ for $\alpha<1$; diverges for $\alpha\geq1$.
• $\int_0^{+\infty} e^{-x},dx = 1$; $\int_0^{+\infty} e^{-x^2},dx = \dfrac{\sqrt{\pi}}{2}$ (Gaussian integral).

Comparison test: if $0\leq f(x)\leq g(x)$ and $\int g$ converges, then $\int f$ converges; if $\int f$ diverges, then $\int g$ diverges.