ON THE DETERMINATION AND THE REDUCTION OF INDEFINITE INTEGRALS IN WHICH THE FUNCTION UNDER THE $$\int $$ SIGN IS THE PRODUCT OF TWO FACTORS EQUAL TO CERTAIN POWERS OF SINES AND OF COSINES OF THE VARIABLE

Dennis M. Cates ()

Chapter Chapter 31 in Cauchy's Calcul Infinitésimal, 2019, pp 167-172 from Springer

Abstract: Abstract Let $$\mu , \nu $$ beIntegration of trigonometric functions two constant quantities, and consider the integral $$\begin{aligned} \int {\sin ^{\mu }{x} \, \cos ^{\nu }{x} \, dx}. \end{aligned}$$ If we set $$\sin ^2{x}=z, $$ or $$\sin {x}=\pm z^{\frac{1}{2}}$$ , this integral will become $$\begin{aligned} \pm \frac{1}{2} \int { z^{\frac{\mu -1}{2}}(1-z)^{\frac{\nu -1}{2}} \, dz}. \end{aligned}$$ Therefore, it can easily be determined (see the twenty-ninth lecture), when the numerical values of the two exponents $$\frac{\mu -1}{2},$$ $$\frac{\nu -1}{2}, $$ and of their sum $$\begin{aligned} \frac{\mu +\nu -2}{2}, \end{aligned}$$ are reduced to three rational numbers, of which one will be an integer number. This is what will necessarily happen whenever the quantities $$\mu , \nu $$ have integer numerical values.

Date: 2019
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DOI: 10.1007/978-3-030-11036-9_31

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