Consider an integral of the type , where R(u,v) is an arbitrary function of two variables defined on the unit circle; we will assume that no trigonometric functions appear in this function itself. Then one can use the universal substitution and obtain an integral which does not feature any trigonometric functions at all. But we also know already that this is a method of highest emergency, so now we will look at conditions that allow us to use simpler substituitons.

1. If for all u, v on the unit circle we have R(-u,v) = -R(u,v), then we can use the substitution y = cos(x).

This in fact means that we consider integrals of the type

where the function G itself does not feature trigonometric functions.

Example:

2. If for all u, v on the unit circle we have R(u,-v) = -R(u,v), then we can use the substitution y = sin(x).

This in fact means that we consider integrals of the type

where the function G itself does not feature trigonometric functions.

Example:

3. If for all u, v on the unit circle we have R(-u,-v) = R(u,v), then we can use the substitution y = tan(x).

This in fact means that we consider integrals of the type

where the function G itself does not feature trigonometric functions.

Example: