Integration Fundamentals

The Integral as Limit of a Sum

The Riemann integral is defined as:

$\int_a^b f(x)\,dx = \lim_{N \to \infty} \sum_{i=1}^{N} f(x_i)\,\delta x$

where $\delta x = (b-a)/N$ and $x_i$ are sample points in each subinterval.

Interpretation: Area under the curve, with rectangles becoming infinitesimally thin.

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Fundamental Theorem of Calculus

First form: If $F'(x) = f(x)$, then:

$\int_a^b f(x)\,dx = F(b) - F(a)$

Second form: Define:

$F(x) = \int_a^x f(t)\,dt$

Then $F'(x) = f(x)$.

Significance: Differentiation and integration are inverse operations.

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Differentiation Under the Integral Sign

For $I(q) = \int_a^b f(x, q)\,dx$:

$\frac{dI}{dq} = \int_a^b \frac{\partial f}{\partial q}\,dx$

If limits depend on $q$:

$\frac{d}{dq}\int_{a(q)}^{b(q)} f(x, q)\,dx = f(b, q)\frac{db}{dq} - f(a, q)\frac{da}{dq} + \int_a^b \frac{\partial f}{\partial q}\,dx$

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Standard Techniques

Substitution

$\int f(g(x))g'(x)\,dx = \int f(u)\,du \quad \text{where } u = g(x)$

Example: $\int 2x\cos(x^2)\,dx$. Let $u = x^2$, $du = 2x\,dx$:

$\int \cos u\,du = \sin u = \sin(x^2)$

Integration by Parts

$\int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx$

Example: $\int x e^x\,dx$.

Let $u = x$, $dv = e^x\,dx$. Then $du = dx$, $v = e^x$:

$\int x e^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x + C$

Partial Fractions

For rational functions, decompose into simpler fractions.

$\frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$

Solving: $1 = A(x+2) + B(x-1)$. At $x = 1$: $A = 1/3$. At $x = -2$: $B = -1/3$.

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Improper Integrals

Type 1: Infinite limits

$\int_a^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx$

Converges if limit exists (finite).

Type 2: Unbounded integrand

$\int_a^b \frac{dx}{\sqrt{x-a}} = \lim_{\varepsilon \to 0^+} \int_{a+\varepsilon}^b \frac{dx}{\sqrt{x-a}}$

Comparison test for improper integrals

If $0 \leq f(x) \leq g(x)$ and $\int g$ converges, then $\int f$ converges.

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Symmetry and Integration

Even function: $f(-x) = f(x)$

$\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx$

Odd function: $f(-x) = -f(x)$

$\int_{-a}^{a} f(x)\,dx = 0$

This simplifies many integrals, especially for trigonometric and polynomial terms.

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Mean Value Theorem for Integrals

For continuous $f$ on $[a, b]$, there exists $c \in [a, b]$ such that:

$\int_a^b f(x)\,dx = f(c)(b - a)$

The integral equals the function value at some point times the interval width.

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Integration by Inspection

Recognise standard forms and their derivatives:

Integral	Result
$\int \frac{f'}{f}\,dx$	$\ln\|f\|$
$\int f'(x)[f(x)]^n\,dx$	$\frac{[f(x)]^{n+1}}{n+1}$
$\int \frac{1}{a^2 + x^2}\,dx$	$\frac{1}{a}\tan^{-1}(x/a)$
$\int \frac{1}{\sqrt{a^2 - x^2}}\,dx$	$\sin^{-1}(x/a)$

The key is identifying patterns that match derivative rules in reverse.