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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:

$√(a^2 - x^2)$ ⇒ $x = a⋅sin(θ)$

$√(x^2 - a^2)$ ⇒ $x = a⋅sec(θ)$

$√(x^2 + a^2)$ ⇒ $x = a⋅tan(θ)$

f(x)F(x)f(x)F(x)
$a$$ax$$x^n$${x^{n+1}}/{n+1}$
$1/x$ln|$x$|$1/{x^n}$${-1}/{(n-1)x^{n-1}}$
$√x$${2/3}x√x$$1/{√x}$$2√x$
$1/{(x-a)(x-b)}$${1/{a-b}}$ln$|{x-a}/{x-b}|$${ax+b}/{cx+d}$${ax}/c-{ad-bc}/{c^2}$ln$|cx+d|$
$1/{x^2+a^2}$${1/a}$arctan$(x/a)$$1/{x^2-a^2}$$1/{2a}$ln$|{x-a}/{x+b}|$
$e^x$$e^x$ln$(x)$$x($ln$(x)-1)$
$a^x$${a^x}/{ln(a)}$log$_a(x)$$x($log$_a(x)-$log$_a(e))$
$xe^{ax}$$1/{a^2}(ax-1)e^{ax}$$x$ln$(ax)$${x^2}/4(2$ln$(ax)-1)$
sin$(x)$-cos$(x)$arcsin$(x)$$x$arcsin$(x)+√{1-x^2}$
cos$(x)$sin$(x)$arccos$(x)$$x$arccos$(x)-√{1-x^2}$
tan$(x)$-ln|cos$(x)$|arctan$(x)$$x$arctan$(x)-1/2$ln$(1+x^2)$
cot$(x)$ln|sin$(x)$|arccot$(x)$$x$arccot$(x)+1/2$ln$(1+x^2)$
sin$^2(x)$$1/2(x-$sin$(x)$cos$(x))$$1/{sin^2(x)}$-cot$(x)$
cos$^2(x)$$1/2(x+$sin$(x)$cos$(x))$$1/{cos^2(x)}$tan$(x)$
tan$^2(x)$tan$(x)-x$$1/{sin(x)}$ln$|{1-cos(x)}/{sin(x)}|$
cot$^2(x)$-cot$(x)-x$$1/{cos(x)}$ln$|{1+sin(x)}/{cos(x)}|$
$1/{1+sin(x)}$${-cos(x)}/{1+sin(x)}$$1/{1-sin(x)}$${cos(x)}/{1-sin(x)}$
$1/{1+cos(x)}$${sin(x)}/{1+cos(x)}$$1/{1-cos(x)}$${-sin(x)}/{1-cos(x)}$
$x$sin$(ax)$$-{1/a}x$cos$(ax)+1/{a^2}$sin$(ax)$$x$cos$(ax)$${1/a}x$sin$(ax)+1/{a^2}$cos$(ax)$
$e^{ax}$sin$(bx)$${e^{ax}}/{a^2+b^2}(a$sin$(bx)-b$cos$(bx))$$e^{ax}$cos$(bx)$${e^{ax}}/{a^2+b^2}(a$cos$(bx)+b$sin$(bx))$
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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