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

$\int \cos (x) d x = \sin (x) + C$

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$\int \sin (x) d x = - \cos (x) + C$

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$\int \tan (x) d x = - \ln | \cos (x) | + C$

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$\int \sec (x) d x = \ln | \sec (x) + \tan (x) | + C$

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$\int \csc (x) d x = - \ln | \csc (x) + \cot (x) | + C$

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$\int \cot (x) d x = \ln | \sin (x) | + C$

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$\int \sec^{2} (x) d x = \tan (x) + C$

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$\int \csc^{2} (x) d x = - \cot (x) + C$

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$\int \sec (x) \tan (x) d x = \sec (x) + C$

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$\int \csc (x) \cot (x) d x = - \csc (x) + C$

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$\int \cos^{2} (x) d x = \frac{1}{2} x + \frac{1}{4} \sin (2 x) + C$

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$\int \sin^{2} (x) d x = \frac{1}{2} x - \frac{1}{4} \sin (2 x) + C$

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$\int \frac{1}{x^{2} + a^{2}} d x = \frac{1}{a} \tan^{- 1} (\frac{x}{a}) + C$

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$\int \frac{1}{\sqrt{a^{2} - x^{2}}} = \sin^{- 1} (\frac{x}{a}) + C$

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$\int \frac{1}{x \sqrt{x^{2} - a^{2}}} = \frac{1}{a} \sec^{- 1} (\frac{| x |}{a}) + C$

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List B.2.4. Integrals Involving Hyperbolic Functions

$\int \cosh (x) d x = \sinh (x) + C$

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$\int \sinh (x) d x = \cosh (x) + C$

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$\int \tanh (x) d x = \ln (\cosh (x)) + C$

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$\int \coth (x) d x = \ln | \sinh (x) | + C$

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$\int \frac{1}{\sqrt{x^{2} - a^{2}}} d x = \ln | x + \sqrt{x^{2} - a^{2}} | + C$

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$\int \frac{1}{\sqrt{x^{2} + a^{2}}} d x = \ln | x + \sqrt{x^{2} + a^{2}} | + C$

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$\int \frac{1}{a^{2} - x^{2}} d x = \frac{1}{2 a} \ln | \frac{a + x}{a - x} | + C$

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

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$\int \frac{1}{x \sqrt{x^{2} + a^{2}}} = \frac{1}{a} \ln | \frac{x}{a + \sqrt{x^{2} + a^{2}}} | + C$

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