Rules of Bioche

The rules of Bioche , in Mathematical, are rules of change of variable in the calculation of Intégrale S comprising of the goniometrical functions. These rules were invented by Bioche when he was professor in special Mathématiques with the Louis-The-Large Lycée. In the continuation, F (T) is a rational expression in sin (T) and cos (T), i.e an expression obtained using sin (T), cos (T), of the real numbers and the four operations $+,\; -,\; \backslash \; times,/$.

Thus, to calculate $\backslash \; int\; F\; (T)\; \backslash \; mathrm\; \{D\}\; t$, one forms $\backslash ,\; \backslash \; Omega\; (T)\; =\; F\; (T)\; \backslash \; mathrm\; \{D\}\; t$. Then,

• If $\backslash \; Omega\; (-\; T)\; =\; \backslash \; Omega\; (T)$, a change of judicious variable is $u\; (T)\; =cos\; (T)$.
• If $\backslash \; Omega\; (\backslash \; pit)\; =\; \backslash \; Omega\; (T)$, a change of judicious variable is $u\; (T)\; =sin\; (T)$.
• If $\backslash \; Omega\; (\backslash \; pi+t)\; =\; \backslash \; Omega\; (T)$, a change of judicious variable is $u\; (T)\; =tan\; (T)$.
• In the other cases, the change of variable $u\; (T)\; =\; \backslash \; mathrm\; \{tan\}\; \backslash \; left\; (\backslash \; frac\; \{T\}\; \{2\}\; \backslash \; right)$ proves often judicious. One will refer on this subject with the article on the trigonometrical formulas implying the tangent of the arc half

These rules do not constitute true a Théorème, but they often lead to the good performance and make it possible if necessary to simplify calculations. They are usable in the majority of the cases only when $f\; (T)$ comprises goniometrical functions. If F is a rational fraction in sin and cos the rules of Bioche always allow to bring back to a primitive rational fraction which is calculated easily by decomposition in simple elements.

Examples of use

• Is the integral $\backslash \; int\; \backslash \; frac\; \{\backslash \; sin\; T\}\; \{1+\; \backslash \; cos^2\; T\}\; \{\backslash \; rm\; D\}\; t$.

$\backslash \; frac\; \{\backslash \; sin\; (-\; T)\}\{1+\; \backslash \; cos^2\; (-\; T)\}\{\backslash \; rm\; D\}\; (-\; T)\; =\; \backslash \; frac\; \{-\; \backslash \; sin\; T\}\; \{1+\; \backslash \; cos^2\; (T)\}(-\; \{\backslash \; rm\; D\}\; T)$ (because $\{\backslash \; rm\; D\}\; (-\; T)\; =\; \{\backslash \; rm\; D\}\; t$ and $\backslash \; sin$ is odd and $\backslash \; cos$ even)

Then according to the rule of Bioche, the best change of variable is $u=\; \backslash \; cos\; t$.

• Is the integral $\backslash \; int\; \backslash \; frac\; \{1\}\; \{\backslash \; cos^2\; T\; (1+\; \backslash \; tan\; T)\}\{\backslash \; rm\; D\}\; t$.

$\backslash \; frac\; \{1\}\; \{(\backslash \; cos^2\; (\backslash \; pi+t))(1+\; \backslash \; tan\; (\backslash \; pi+t))\}\{\backslash \; rm\; D\}\; (\backslash \; pi+t)\; =\; \backslash \; frac\; \{1\}\; \{(\backslash \; cos^2\; T)\; (1+\; \backslash \; tan\; T)\}\{\backslash \; rm\; D\}\; t$ (because $\{\backslash \; rm\; D\}\; (\backslash \; pi+t)\; =\; \{\backslash \; rm\; D\}\; t$ and $\backslash \; cos\; (\backslash \; pi+t)\; =\; \backslash \; cos\; t$ and $\backslash \; tan\; (\backslash \; pi+t)\; =\; \backslash \; tan\; t$)

Then according to the rule of Bioche, the most suitable change of variable is $u=\; \backslash \; tan\; t$.

Once the change of variable carried out, these two integrals can be calculated more easily because they comprise functions which one knows primitiver.

Another version: hyperbolic functions

That is to say to calculate $\backslash \; int\; \{G\; (\backslash \; cosh\; (T),\; \backslash \; sinh\; (T))\backslash ,\; \backslash \; textrm\; \{D\}\; T\}$. If the rules of Bioche suggest calculating $\backslash \; int\; \{G\; (\backslash \; cos\; (T),\; \backslash \; sin\; (T))\backslash ,\; \backslash \; textrm\; \{D\}\; T\}$ by u=cos (T) (resp. sin (T), tan (T), tan (t/2)) a change of judicious variable for the first integral is u=cosh (T) (resp. sinh (T), tanh (T), tanh (t/2)). In all the cases, the change of variable $u=e^\; \{T\}$ makes it possible to bring back to a primitive rational fraction, this last change of variable being more interesting in the fourth case (u=tanh (t/2)).

See too

• Integral calculus

• numerical Calculation of an integral
• Integration by change of variable

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