
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$(x/r)$ |
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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.
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