Weierstrass substitution


In integral calculus, the Weierstrass substitution or tangent half-angle substitution is a method for evaluating integrals, which converts a rational function of trigonometric functions of into an ordinary rational function of by setting. No generality is lost by taking these to be rational functions of the sine and cosine. The general transformation formula is
It is named after Karl Weierstrass, though it can be found in a book by Leonhard Euler from 1768. Michael Spivak wrote that this method was the "sneakiest substitution" in the world.

The substitution

Starting with a rational function of sines and cosines, one replaces and with rational functions of the variable and relates the differentials and as follows.
Let, where. Then
Hence,

Derivation of the formulas

By the double-angle formulas,
and
Finally, since,

Examples

First example: the cosecant integral

We can confirm the above result using a standard method of evaluating the cosecant integral by multiplying the numerator and denominator by and performing the following substitutions to the resulting expression: and. This substitution can be obtained from the difference of the derivatives of cosecant and cotangent, which have cosecant as a common factor.
Now, the half-angle formulas for sines and cosines are
They give
so the two answers are equivalent. Alternatively, one can use a tangent half-angle identity to get
The secant integral may be evaluated in a similar manner.

Second example: a definite integral

In the first line, one does not simply substitute for both limits of integration. The singularity of at must be taken into account. Alternatively, first evaluate the indefinite integral then apply the boundary values.
By symmetry,
which is the same as the previous answer.

Third example

If

Geometry

As x varies, the point winds repeatedly around the unit circle centered at . The point
goes only once around the circle as t goes from −∞ to +∞, and never reaches the point , which is approached as a limit as t approaches ±∞. As t goes from −∞ to −1, the point determined by t goes through the part of the circle in the third quadrant, from to . As t goes from −1 to 0, the point follows the part of the circle in the fourth quadrant from to . As t goes from 0 to 1, the point follows the part of the circle in the first quadrant from to . Finally, as t goes from 1 to +∞, the point follows the part of the circle in the second quadrant from to .
Here is another geometric point of view. Draw the unit circle, and let P be the point. A line through P is determined by its slope. Furthermore, each of the lines intersects the unit circle in exactly two points, one of which is P. This determines a function from points on the unit circle to slopes. The trigonometric functions determine a function from angles to points on the unit circle, and by combining these two functions we have a function from angles to slopes.

Gallery

Hyperbolic functions

As with other properties shared between the trigonometric functions and the hyperbolic functions, it is possible to use hyperbolic identities to construct a similar form of the substitution: