Bioche's rules

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Bioche's rules, formulated by the French mathematician Charles Bioche [fr] (1859–1949), are rules to aid in the computation of certain indefinite integrals in which the integrand contains sines and cosines.

In the following, ${\displaystyle f(t)}$ is a rational expression in ${\displaystyle \sin t}$ and ${\displaystyle \cos t}$. In order to calculate ${\displaystyle \int f(t)\,dt}$, consider the integrand ${\displaystyle \omega (t)=f(t)\,dt}$. We consider the behavior of this entire integrand, including the ${\textstyle dt}$, under translation and reflections of the t axis. The translations and reflections are ones that correspond to the symmetries and periodicities of the basic trigonometric functions.

Bioche's rules state that:

1. If ${\displaystyle \omega (-t)=\omega (t)}$, a good change of variables is ${\displaystyle u=\cos t}$.
2. If ${\displaystyle \omega (\pi -t)=\omega (t)}$, a good change of variables is ${\displaystyle u=\sin t}$.
3. If ${\displaystyle \omega (\pi +t)=\omega (t)}$, a good change of variables is ${\displaystyle u=\tan t}$.
4. If two of the preceding relations both hold, a good change of variables is ${\displaystyle u=\cos 2t}$.
5. In all other cases, use ${\displaystyle u=\tan(t/2)}$.

Because rules 1 and 2 involve flipping the t axis, they flip the sign of dt, and therefore the behavior of ω under these transformations differs from that of ƒ by a sign. Although the rules could be stated in terms of ƒ, stating them in terms of ω has a mnemonic advantage, which is that we choose the change of variables u(t) that has the same symmetry as ω.

These rules can be, in fact, stated as a theorem: one shows^[1] that the proposed change of variable reduces (if the rule applies and if f is actually of the form ${\displaystyle f(t)={\frac {P(\sin t,\cos t)}{Q(\sin t,\cos t)}}}$) to the integration of a rational function in a new variable, which can be calculated by partial fraction decomposition.