Tangent half-angle substitution

The Weierstrass substitution is a method from the mathematical subfield of Analysis. It is a variant of integration by substitution, which can be applied to specific integrand with trigonometric functions. Is named the method after the mathematician Karl Weierstrass, who developed it.

Description of the substitution

Be two real numbers and a rational function. To an integral of the form

To calculate, substitution

Be applied. For the functions sine and cosine are then obtained the substitutions

And applies to the differential

Since the functions tangent, cotangent, secant and cosecant than can be written as fractions with sine and cosine, the Weierstrass substitution can also be applied to these trigonometric functions. Are the substitutions

Example

General substitution is suitable to eliminate the trigonometric functions in the calculation of the integral, as the following example shows.

This integral can now be calculated with a further integration by substitution.

Derivation

In this section, the substitution formulas for sine and cosine are derived. With the addition theorems we obtain:

Together we have the presentation up for. The representation for is obtained as follows:

The derivative of x with respect to t is given by: