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Trigonometric integration

In engineering, rotational action is often translated into lineal action, or vice-versa. Sine, cosine, and tangent, are three trigonometric functions which describe lateral and transverse displacements, and their ratio, as a radius rotates through the circle it describes.

Basic trig identities
Mathematics question

Summary of Integral Properties and Solutions

$∫sin x dx = -cos x + C$

$∫cos x dx = sin x + C$

$∫sin (ax + b) dx = -1/{a}cos (ax + b) + C$

$∫cos (ax + b) dx = 1/{a}sin(ax + b) + C$

Trigonometric substitutions

If the integrand contains a quadratic radical expression, these trig substitutions may be used:

$√(a^2 - x^2)$ ⇒ $x = a⋅sin(θ)$

$√(x^2 - a^2)$ ⇒ $x = a⋅sec(θ)$

$√(x^2 + a^2)$ ⇒ $x = a⋅tan(θ)$

sinh$(x)$cosh$(x)$arsinh$(x)$$x$arsinh$(x) - √{x^2+1}$
cosh$(x)$sinh$(x)$arcosh$(x)$$x$arcosh$(x) - √{x^2-1}$
tanh$(x)$ln(cosh$(x)$)artanh$(x)$$x$artanh$(x) +1/2$ln$(1-x^2)$
coth$(x)$ln|sinh$(x)$|arcoth$(x)$$x$arcoth$(x) +1/2$ln$(x^2-1)$
$√{x^2+a}$$1/2x√{x^2+a} + a/2$ln$|x+√{x^2+a}|$$1/{√{x^2+a}}$ln$|x+√{x^2+a}|$
$√{r^2-x^2}$$1/2x√{x^2-x^2} + {r^2}/2$arcsin$(x/r)$$1/{√{r^2-x^2}}$arcsin$(x/r)$

Content © Andrew Bone. All rights reserved. Created : February 5, 2015

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