LESSONS 29-34. _Integration of Functions of a Single Variable._ Object of the integral calculus. There always exists a function which has a given function for its derivative. Indefinite integrals. Definite integrals. Notation. Integration by separation, by substitution, by parts. Integration of rational differentials, integer or fractional, in the several cases which may present themselves. Integration of the algebraical differentials, which contain a radical of the second degree of the form √(_c+bx+ax^{2}_). Different transformations which render the differential rational. Reduction of the radical to one of the forms √(x^{2}+x^{2}), √(a^{2}-x^{2}), √(x^{2}-a^{2}). Integration of the algebraical differentials which contain two radicals of the form √(a+x), √(b+x), or any number of monomials affected with fractional indices. Application to the expressions x^{m} dx dx x^{m} dx ---------- , ---------------- , -------- √(1-x^{2}) x^{m} √(1-x^{2}) √(ax-x) Integration of the differentials dx dx F(log x)-- , F sin^{-1}x ---------- , x √(1-x^{2}) x(log x^{n})dx, x^{m} e^{ax}dx, (sin^{-1}x^{m})dx. Integration of the differentials e^{ax} sin _bxdx_ and e^{ax} cos _bxdx_. Integration of (sin x^{m}.)(cos x^{n}) _dx_. Integration by series. Application to the expression dx ------------------- √(ax-x^{2}) √(1-bx) Application of integration by series to the development of functions, the development of whose derivatives is given: tan^{-1}_x_, sin^{-1}_x_, log(1 + _x_.) LESSONS 35-38. _Geometrical Applications._ Quadrature of certain curves. Circle, hyperbola, cycloid, logarithmic spiral, &c. Rectification of curves by rectilinear or polar co-ordinates. Examples. Numerical applications. Cubic content of solids of revolution. Quadrature of their surfaces. Cubic content of solids in general, with rectilinear or polar co-ordinates. Numerical applications. Quadrature of any curved surfaces expressed by rectangular co-ordinates. Application to the sphere. LESSONS 39-42. _Mechanical Applications._ General formula for the determination of the center of gravity of solids, curved or plane surfaces, and arcs of curves. Various applications. Guldin’s theorem. Volume of the truncated cylinder. General formula which represent the components of the attraction of a body upon a material point, upon the supposition that the action upon each element varies inversely as the square of the distance. Attraction of a spherical shell on an external or internal point.