The Weierstrass substitution formulas for $-\pi < x < \pi$ are:

They can be obtained in the following manner:

Make the Weierstrass substitution $\displaystyle t=\tan \left( \frac{x}{2} \right)$ . (This substitution is also known as the universal trigonometric substitution.) Then we have

and

Note that these are just the ``formulas involving radicals'' as designated in the entry goniometric formulas; however, due to the restriction on $x$ , the $\pm$ 's are unnecessary.

Using the above formulas along with the double angle formulas, we obtain

and

Finally, since $\displaystyle t=\tan\left( \frac{x}{2} \right)$ , solving for $x$ yields that $x=2\arctan t$ . Thus, $\displaystyle dx=\frac{2}{1+t^2} \, dt$ .

The Weierstrass substitution formulas are most useful for integrating rational functions of sine and cosine.