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A.1. Basic integration formulas

A.1.1 Fundamental techniques#

Technique	Formula
Integration by parts	$\int u d v = uv - \int v d u$

A.1.2 Power rule and elementary functions

\int u^{n} d u = \frac{1}{n + 1} u^{n + 1} + C, n \neq = - 1

$n \neq = - 1$

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\int \frac{d u}{u} = ln ∣ u ∣ + C

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\int e^{u} d u = e^{u} + C

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\int a^{u} d u = \frac{a ^{u}}{ln a} + C, a > 0, a \neq = 1

$a > 0, a \neq = 1$

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A.1.3 Basic trigonometric integrals

\int sin u d u = - cos u + C

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\int cos u d u = sin u + C

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\int sec^{2} u d u = tan u + C

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\int csc^{2} u d u = - cot u + C

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\int sec u tan u d u = sec u + C

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\int csc u cot u d u = - csc u + C

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A.1.4 Trigonometric integrals (logarithmic forms)

\int tan u d u = - ln ∣ cos u ∣ + C

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\int cot u d u = ln ∣ sin u ∣ + C

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\int sec u d u = ln ∣ sec u + tan u ∣ + C

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\int csc u d u = ln ∣ csc u - cot u ∣ + C

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A.1.5 Integrals yielding inverse trigonometric functions

\int \frac{d u}{a ^{2} - u ^{2}} = arcsin \frac{u}{a} + C

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\int \frac{d u}{a ^{2} + u ^{2}} = \frac{1}{a} arctan \frac{u}{a} + C

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\int \frac{d u}{u u ^{2} - a ^{2}} = \frac{1}{a} arcsec \frac{u}{a} + C

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A.1.6 Partial fractions (logarithmic forms)

\int \frac{d u}{a ^{2} - u ^{2}} = \frac{1}{2 a} ln \frac{u + a}{u - a} + C

$ln \frac{u + a}{u - a} + C$

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\int \frac{d u}{u ^{2} - a ^{2}} = \frac{1}{2 a} ln \frac{u - a}{u + a} + C

$ln \frac{u - a}{u + a} + C$

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