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

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.