Tangent half-angle substitution

Summary

The tangent half-angle substitution sets $t = \tan(x/2)$ in an integral. It turns each rational expression in $\sin x$ and $\cos x$ into a rational function of $t$. Partial fractions then finish the computation.

Remarks

The substitution gives $\sin x = \frac{2t}{1 + t^2}$, $\cos x = \frac{1 - t^2}{1 + t^2}$ and $dx = \frac{2 \, dt}{1 + t^2}$. It works on intervals where $x/2$ stays inside $(-\pi/2, \pi/2)$. For example, it gives $\int \frac{dx}{\sin x} = \ln |\tan(x/2)| + C$.

Explanations

L1 human

This substitution turns trigonometric integrals into integrals of fractions of polynomials. Those integrals are always solvable by partial fractions.