Weierstrass Substitution

The Weierstrass substitution, named after German mathematician Karl Weierstrass (1815−1897), is used for converting rational expressions of trigonometric functions into algebraic rational functions, which may be easier to integrate.

This method of integration is also called the tangent half-angle substitution as it implies the following half-angle identities:

where $$ or $$

The differential $$ is determined as follows:

Any rational expression of trigonometric functions can be always reduced to integrating a rational function by making the Weierstrass substitution.

The Weierstrass substitution is very useful for integrals involving a simple rational expression in $$ and/or $$ in the denominator.

To calculate an integral of the form $$ where $$ is a rational function, use the substitution $$

Similarly, to calculate an integral of the form $$ where $$ is a rational function, use the substitution $$

If an integrand is a function of only $$ the substitution $$ converts this integral into integral of a rational function.

To calculate an integral of the form $$ where both functions $$ and $$ have even powers, use the substitution $$ and the formulas

Solved Problems

Solution.

We use the universal trigonometric substitution:

Since $$ we have

Solution.

Using the Weierstrass substitution

we can rewrite the integral in the form

Complete the square in the denominator:

Changing $$ $$ gives the final answer:

Solution.

Make the universal trigonometric substitution:

Then the integral becomes

Solution.

Using the substitution

we can easily find the integral:we can easily find the integral:

Solution.

To simplify the integral, we use the Weierstrass substitution:

The integral becomes

Solution.

As in the previous examples, we will use the universal trigonometric substitution:

Since $$ $$ we can write:

Solution.

Making the $$ substitution, we have

Then the integral in $$terms is written as

Solution.

We can write the integral in form:

Use the universal trigonometric substitution:

This leads to the following result: