The simpler methods of integration

Integration of the product of a constant term and a differential. On page 38, it was pointed out that the differential of the product of a variable and a constant, is equal to the constant multiplied by the differential of the variable . It follows directly that the integral of the product of a constant and a differential, is equal to the constant multiplied by the integral of the differential. E.g., if a is constant, 

On the other hand, the value of an integral is altered if a term containing one of the variables is placed outside the integral sign. For instance, the reader will see very shortly that while xHx = xjxdx = xz. 

A constant term must he added to every integral. It has been shown that a constant term always disappears from an expression during differentiation, thus, 

This is equivalent to stating that there is an infinite number of expressions, differing only in the value of the constant term, which, when differentiated, produce the same differential. In 

by slipping in another simplifying assumption Clairaut expressed his 

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