12. The processes of the integral calculus consist largely in transformations Indefinite Integrals. of the functions to be integrated into such forms that they can be recognized as differential coefficients of functions which have previously been differentiated. Corresponding to the results in the table of § 11 we have those in the following table:—
| ƒ(x) | ∫ƒ(x)dx |
| xn | xn+1 / (n + 1) for all values of n except −1 |
| 1/x | loge x |
| eax | a−1eax |
| cos x | sin x |
| sin x | −cos x |
| (a2 − x2)−1/2 | sin−1 (x/a) |
| 1 / (a2 + x2) | (1/a) tan−1 (x/a) |
The formal rules of § 11 give us means for the transformation of integrals into recognizable forms. For example, the rule (ii.) for a sum leads to the result that the integral of a sum of a finite number of terms is the sum of the integrals of the several terms. The rule (iii.) for a product leads to the method of integration by parts. The rule (v.) for a function of a function leads to the method of substitution (see § 48 below.)
II. History.
13. The new limiting processes which were introduced in the development of the higher analysis were in the first instance related to problems of the integral calculus. Johannes Kepler in his Astronomia nova ... de motibus stellae Martis Kepler’s methods of Integration. (1609) stated his laws of planetary motion, to the effect that the orbits of the planets are ellipses with the sun at a focus, and that the radii vectores drawn from the sun to the planets describe equal areas in equal times. From these statements it is to be concluded that Kepler could measure the areas of focal sectors of an ellipse. When he made out these laws there was no method of evaluating areas except the Greek methods. These methods would have sufficed for the purpose, but Kepler invented his own method. He regarded the area as measured by the “sum of the radii” drawn from the focus, and he verified his laws of planetary motion by actually measuring a large number of radii of the orbit, spaced according to a rule, and adding their lengths.
| Fig. 5. |
He had observed that the focal radius vector SP (fig. 5) is equal to the perpendicular SZ drawn from S to the tangent at p to the auxiliary circle, and he had further established the theorem which we should now express in the form—the differential element of the area ASp as Sp turns about S, is equal to the product of SZ and the differential adφ, where a is the radius of the auxiliary circle, and φ is the angle ACp, that is the eccentric angle of P on the ellipse. The area ASP bears to the area ASp the ratio of the minor to the major axis, a result known to Archimedes. Thus Kepler’s radii are spaced according to the rule that the eccentric angles of their ends are equidifferent, and his “sum of radii” is proportional to the expression which we should now write
∫φ0 (a + ae cos φ) dφ,
where e is the eccentricity. Kepler evaluated the sum as proportional to φ + e sin φ.