On Indefinite Integrals Which Contain Exponential, Logarithmic, Or Circular Functions

We call exponential functions or logarithmic functions those which contain variable exponents or logarithms, and trigonometric or circular functions those which contain trigonometric lines or arcs (Recall the phrase “trigonometric lines” is an old term used to denote the signed lengths of the line segments represented by the six trigonometric functions. Picturing a unit circle with a particular signed angle along with its corresponding terminal side, the line segments referred to here are those generated if one were to graphically construct the trigonometric function representations on the corresponding right triangle about this circle. Today, we would simply refer to “trigonometric lines” as the values of the trigonometric functions of a particular angle. The term “arcs” is another older word referring to the values of the inverse trigonometric functions, or the signed angle mentioned earlier. In the case of a unit circle, this coincides with the signed length of the arc.) of a circle. It would be very useful to integrate the differential formulas which contain similar functions. But, we do not have sure methods to achieve this, except in a small number of particular cases that we will now review.