Integral Calculus Notes

Last reviewed 16 Sept 2026 · 6 min read

Indefinite integrals

$\int f (x) d x = F (x) + C$ where $F^{'} (x) = f (x)$ .

Standard integrals

$f (x)$	$\int f (x) d x$
$x^{n}$ ( $n \neq = - 1$ )	$\frac{x ^{n + 1}}{n + 1}$
$\frac{1}{x}$	$ln ∣ x ∣$
$e^{a x}$	$\frac{e ^{a x}}{a}$
$a^{x}$	$\frac{a ^{x}}{ln a}$
$sin x$ / $cos x$	$- cos x$ / $sin x$
$sec^{2} x$ / $csc^{2} x$	$tan x$ / $- cot x$
$tan x$	$ln ∣ sec x ∣$
$sec x$	$ln ∣ sec x + tan x ∣$
$\frac{1}{a ^{2} + x ^{2}}$	$\frac{1}{a} tan^{- 1} \frac{x}{a}$
$\frac{1}{a ^{2} - x ^{2}}$	$sin^{- 1} \frac{x}{a}$
$\frac{1}{x ^{2} - a ^{2}}$	$\frac{1}{2 a} ln \frac{x - a}{x + a}$
$\frac{1}{x ^{2} + a ^{2}}$	$ln x + x^{2} + a^{2}$
$a^{2} - x^{2}$	$\frac{x}{2} a^{2} - x^{2} + \frac{a ^{2}}{2} sin^{- 1} \frac{x}{a}$
$e^{a x} sin b x$	$\frac{e ^{a x}}{a ^{2} + b ^{2}} (a sin b x - b cos b x)$

Methods of integration

Substitution

Replace a function and its derivative: $\int f (g (x)) g^{'} (x) d x = \int f (u) d u$ . In particular $\int \frac{f ^{'} ( x )}{f ( x )} d x = ln ∣ f (x) ∣$ .

Integration by parts

\int u v d x = u \int v d x - \int (u^{'} \int v d x) d x

Choose $u$ in the order ILATE — Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential.

Partial fractions

Rational functions $\frac{P ( x )}{Q ( x )}$ (proper) are split: linear factors → $\frac{A}{x - a}$ ; repeated → $\frac{A}{x - a} + \frac{B}{( x - a ) ^{2}}$ ; quadratic factors → $\frac{A x + B}{x ^{2} + p x + q}$ .

Trigonometric integrals

Use identities: $sin^{2} x = \frac{1 - cos 2 x}{2}$ , $cos^{2} x = \frac{1 + cos 2 x}{2}$ , product-to-sum formulas, and substitutions $t = tan (x /2)$ for rational functions of $sin x$ , $cos x$ .

Definite integrals

Fundamental theorem of calculus: $\int_{a}^{b} f (x) d x = F (b) - F (a)$ .

Properties

$\int_{a}^{b} f d x = - \int_{b}^{a} f d x$ ; $\int_{a}^{a} f d x = 0$ .
$\int_{a}^{b} f d x = \int_{a}^{c} f d x + \int_{c}^{b} f d x$ .
$\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$ — very useful.
$\int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ if $f$ is even; $= 0$ if $f$ is odd.
$\int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ if $f (2 a - x) = f (x)$ ; $= 0$ if $f (2 a - x) = - f (x)$ .
For a periodic function with period $T$ : $\int_{0}^{n T} f d x = n \int_{0}^{T} f d x$ .

Wallis' formulas

FormulaWallis' formulas

\int_{0}^{π /2} sin^{n} x d x = \int_{0}^{π /2} cos^{n} x d x = \frac{( n - 1 ) ( n - 3 ) \dots}{n ( n - 2 ) \dots} \times k

$k = \frac{π}{2}$ if $n$ is even; $k = 1$ if $n$ is odd.

\int_{0}^{π /2} sin^{m} x cos^{n} x d x = \frac{[( m - 1 ) ( m - 3 ) \dots ] [( n - 1 ) ( n - 3 ) \dots ]}{( m + n ) ( m + n - 2 ) \dots} \times k

$k = \frac{π}{2}$ if both $m$ and $n$ are even; otherwise $k = 1$ .

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Integral Calculus