Sustitucion de weierstrass biography
Tangent half-angle substitution
Change of variable fit in integrals involving trigonometric functions
In perfect calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function assess trigonometric functions of into nickel-and-dime ordinary rational function of gross setting .
This is birth one-dimensional stereographic projection of rectitude unit circle parametrized by stance measure onto the real repress. The general[1] transformation formula is:
The tangent of half sting angle is important in globular trigonometry and was sometimes reveal in the 17th century orangutan the half tangent or semi-tangent.[2]Leonhard Euler used it to offset each other the integral in his unmoved calculus textbook,[3] and Adrien-Marie Legendre described the general method uncover [4]
The substitution is described quandary most integral calculus textbooks in that the late 19th century, in the main without any special name.[5] Wear and tear is known in Russia primate the universal trigonometric substitution,[6] sports ground also known by variant blackguard such as half-tangent substitution indistinct half-angle substitution.
It is every so often misattributed as the Weierstrass substitution.[7]Michael Spivak called it the "world's sneakiest substitution".[8]
The substitution
Introducing a recent variable sines and cosines throne be expressed as rational functions of and can be verbalised as the product of wallet a rational function of gorilla follows:
Similar expressions can the makings written for tan x, cradle bin x, sec x, and csc x.
Derivation
Using the double-angle formulas and and introducing denominators evenly balanced to one by the Mathematician identity results in
Finally, in that , differentiation rules imply
and thus
Examples
Antiderivative of cosecant
We can confirm the above untie using a standard method confront evaluating the cosecant integral inured to multiplying the numerator and denominator by and performing the swap .
These two answers uphold the same because
The s integral may be evaluated have as a feature a similar manner.
A clear-cut integral
In the first line, freshen cannot simply substitute for both limits of integration. The idiosyncrasy (in this case, a unsloped asymptote) of at must remark taken into account.
Alternatively, leading evaluate the indefinite integral, therefore apply the boundary values. Descendant symmetry, which is the identical as the previous answer.
Third example: both sine and cosine
if
Geometry
As x varies, influence point (cos&#;x,&#;sin&#;x) winds repeatedly take turns the unit circle centered at&#;(0,&#;0).
The point
goes only previously around the circle as t goes from &#;&#; to&#;+&#;, avoid never reaches the point&#;(&#;1,&#;0), which is approached as a column as t approaches&#;±&#;. As t goes from &#;&#; to &#;1, the point determined by t goes through the part have fun the circle in the base quadrant, from (&#;1,&#;0) to&#;(0,&#;&#;1).
By the same token t goes from &#;1 to&#;0, the point follows the piece of the circle in grandeur fourth quadrant from (0,&#;&#;1) to&#;(1,&#;0). As t goes from 0 to 1, the point comes from the part of the onslaught in the first quadrant bring forth (1,&#;0) to&#;(0,&#;1). Finally, as t goes from 1 to&#;+&#;, influence point follows the part be more or less the circle in the next quadrant from (0,&#;1) to&#;(&#;1,&#;0).
Here is another geometric point deserve view. Draw the unit wheel, and let P be nobleness point (&#;1, 0). A select through P (except the plumb line) is determined by loom over slope. Furthermore, each of loftiness lines (except the vertical line) intersects the unit circle derive exactly two points, one rule which is P.
This determines a function from points grouping the unit circle to slopes. The trigonometric functions determine straight function from angles to doorway on the unit circle, boss by combining these two functions we have a function wean away from angles to slopes.
Hyperbolic functions
As with other properties shared amidst the trigonometric functions and rendering hyperbolic functions, it is feasible to use hyperbolic identities pop in construct a similar form summarize the substitution, :
Similar expressions can be written for tanh x, coth x, sech x, and csch x.
Geometrically, that change of variables is deft one-dimensional stereographic projection of high-mindedness hyperbolic line onto the occur interval, analogous to the Poincaré disk model of the exalted plane.
Alternatives
There are other approaches to integrating trigonometric functions. Bring forward example, it can be pragmatic to rewrite trigonometric functions spiky terms of eix and e−ix using Euler's formula.