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$(x/r)$ |
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"I knew Descartes," said Isaac Barrow. "René', I used to say. 'Have you got des cartes?' Then after a few tankards, he would say: 'Don't try to cheat me, Wheelie-boy. Cogito ergo sum...' - that's classical pidgin for 'I can think so I can add up as well as the next man'."
