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# 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.

Chapter

## 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.

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© Springer Nature Switzerland AG 2019

## Authors and Affiliations

1. 1.Sun CityUSA