Bioche's Rules
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Bioche's rules, formulated by the French mathematician (1859–1949), are rules to aid in the computation of certain
indefinite integral In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
s in which the
integrand In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
contains
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
s and cosines. In the following, f(t) is a rational expression in \sin t and \cos t. In order to calculate \int f(t)\,dt, consider the integrand \omega(t)=f(t)\,dt. We consider the behavior of this entire integrand, including the theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
: one shows that the proposed change of variable reduces (if the rule applies and if ''f'' is actually of the form f(t) = \frac) to the integration of a
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
in a new variable, which can be calculated by
partial fraction decomposition In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
.


Case of polynomials

To calculate the integral \int\sin^p(t)\cos^q(t)dt, Bioche's rules apply as well. * If ''p'' and ''q'' are odd, one uses u = \cos(2t); * If ''p'' is odd and ''q'' even, one uses u = \cos(t); * If ''p'' is even and ''q'' odd, one uses u = \sin(t); * If not, one is reduced to lineariz.


Another version for hyperbolic functions

Suppose one is calculating \int g(\cosh t, \sinh t)dt. If Bioche's rules suggest calculating \int g(\cos t, \sin t)dt by u = \cos(t) (respectively, \sin t, \tan t, \cos(2t), \tan(t/2)), in the case of hyperbolic sine and cosine, a good change of variable is u = \cosh(t) (respectively, \sinh(t), \tanh(t), \cosh(2t), \tanh(t/2)). In every case, the change of variable u = e^t allows one to reduce to a rational function, this last change of variable being most interesting in the fourth case (u = \tanh(t/2)).


Examples


Example 1

As a trivial example, consider ::\int \sin t \,dt. Then f(t)=\sin t is an odd function, but under a reflection of the ''t'' axis about the origin, ω stays the same. That is, ω acts like an even function. This is the same as the symmetry of the cosine, which is an even function, so the mnemonic tells us to use the substitution u=\cos t (rule 1). Under this substitution, the integral becomes -\int du. The integrand involving transcendental functions has been reduced to one involving a rational function (a constant). The result is -u+c=-\cos t+c, which is of course elementary and could have been done without Bioche's rules.


Example 2

The integrand in ::\int \frac has the same symmetries as the one in example 1, so we use the same substitution u=\cos t. So ::\frac = - \frac = - \frac. This transforms the integral into ::\int - \frac, which can be integrated using partial fractions, since \frac = \frac \left( \frac+\frac\right). The result is that ::\int \frac=-\frac\ln\frac+c.


Example 3

Consider ::\int \frac, where \beta^2<1. Although the function ''f'' is even, the integrand as a whole ω is odd, so it does not fall under rule 1. It also lacks the symmetries described in rules 2 and 3, so we fall back to the last-resort substitution u=\tan(t/2). Using \cos t=\frac and a second substitution v=\sqrtu leads to the result ::\int \frac = \frac\arctan\left frac\right+ c.


References

{{Reflist * Zwillinger, ''Handbook of Integration'', p. 108 * Stewart, ''How to Integrate It: A practical guide to finding elementary integrals'', pp. 190−197. Integral calculus Theorems in calculus