APEX Integration Formulas

Section B.2 Integration Formulas

List B.2.1. Basic Rules

List B.2.2. Integrals of Elementary (non-Trig) Functions

List B.2.3. Integrals Involving Trigonometric Functions

$\int \cos x d x = \sin x + C$
$\int \sin x d x = - \cos x + C$
$\int \tan x d x = - \ln | \cos x | + C$
$\int \sec x d x = \ln | \sec x + \tan x | + C$
$\int \csc x d x = - \ln | \csc x + \cot x | + C$
$\int \cot x d x = \ln | \sin x | + C$
$\int \sec^{2} x d x = \tan x + C$
$\int \csc^{2} x d x = - \cot x + C$
$\int \sec x \tan x d x = \sec x + C$
$\int \csc x \cot x d x = - \csc x + C$
$\int \cos^{2} x d x = \frac{1}{2} x + \frac{1}{4} \sin (2 x) + C$
$\int \sin^{2} x d x = \frac{1}{2} x - \frac{1}{4} \sin (2 x) + C$
$\int \frac{1}{x^{2} + a^{2}} d x = \frac{1}{a} \tan^{- 1} (\frac{x}{a}) + C$
$\int \frac{1}{\sqrt{a^{2} - x^{2}}} = \sin^{- 1} (\frac{x}{a}) + C$
$\int \frac{1}{x \sqrt{x^{2} - a^{2}}} = \frac{1}{a} \sec^{- 1} (\frac{| x |}{a}) + C$

List B.2.4. Integrals Involving Hyperbolic Functions

$\int \cosh x d x = \sinh x + C$
$\int \sinh x d x = \cosh x + C$
$\int \tanh x d x = \ln (\cosh x) + C$
$\int \coth x d x = \ln | \sinh x | + C$
$\int \frac{1}{\sqrt{x^{2} - a^{2}}} d x = \ln | x + \sqrt{x^{2} - a^{2}} | + C$
$\int \frac{1}{\sqrt{x^{2} + a^{2}}} d x = \ln | x + \sqrt{x^{2} + a^{2}} | + C$
$\int \frac{1}{a^{2} - x^{2}} d x = \frac{1}{2 a} \ln | \frac{a + x}{a - x} | + C$
$\int \frac{1}{x \sqrt{a^{2} - x^{2}}} d x = \frac{1}{a} \ln (\frac{x}{a + \sqrt{a^{2} - x^{2}}}) + C$
$\int \frac{1}{x \sqrt{x^{2} + a^{2}}} = \frac{1}{a} \ln | \frac{x}{a + \sqrt{x^{2} + a^{2}}} | + C$

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