
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.
$∫sin x dx = -cos x + C$
$∫cos x dx = sin x + C$
$∫sin (ax + b) dx = -1/{a}cos (ax + b) + C$
$∫cos (ax + b) dx = 1/{a}sin(ax + b) + C$
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 © Renewable-Media.com. All rights reserved. Created : February 5, 2015
The most recent article is:
View this item in the topic:
and many more articles in the subject:
Environmental Science is the most important of all sciences. As the world enters a phase of climate change, unprecedented biodiversity loss, pollution and human population growth, the management of our environment is vital for our futures. Learn about Environmental Science on ScienceLibrary.info.
1454 - 1519
Leonardo da Vinci's name is synonymous with genius and polymath. The range of his interests and talents seems endless. Despite his many lines of scientific investigation, his work in anatomy and botany was largely lost till rediscovered centuries later.
Website © renewable-media.com | Designed by: Andrew Bone