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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)$

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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.

Johann Bernoulli (III), 1744 - 1807