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.

Abstract

Let $\mu , \nu$ be 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.