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.

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$(x/r)$

Site Index

Latest Item on Science Library:

The most recent article is:

Air Resistance and Terminal Velocity

View this item in the topic:

Mechanics

and many more articles in the subject:

Physics

Subject of the Week

Resources

Science resources on ScienceLibrary.info. Games, puzzles, enigmas, internet resources, science fiction and fact, the weird and the wonderful things about the natural world. Have fun while learning Science with ScienceLibrary.info.

Great Scientists

Johann(III) Bernoulli

1744 - 1807

Johann Bernoulli (III) lived and worked in Berlin, where he was director of the Mathematics Department of the Academy of Berlin, and the last noted mathematician of the Bernoulli dynasty of mathematicians.