Differentiation of a definite integral with respect to a parameter in it, which is made to vary. Geometrical demonstration of the formula. Integration under the sign of integration. Application to the determination of certain definite integrals.
Determination of the integrals ∫ sin ax x dx, and ∫ cos bx sin ax x dx, between the limits 0 and x. Remarkable discontinuity which these integrals present.
Determination of ∫e-x²dx and ∫e-x²cos mx dx between the limits 0 and ∞.
Lesson 3. Integration of Differentials containing several Variables.
Condition that an expression of the form M dx + N dy in which M and N are given functions of x and y may be an exact differential of two independent variables x and y. When this condition is satisfied, to find the function.
Extension of this theory to the case of three variables.
Lessons 4–6. Integration of Differential Equations of the First Order.
Differential equations of the first order with two variables. Problem in geometry to which these equations correspond. What is meant by their integral. This integral always exists, and its expression contains an arbitrary constant.
Integration of the equation M dx + N dy = 0 when its first member is an exact differential. Whatever the functions M and N may be there always exists a factor µ, such that µ (M dx + N dy) is an exact differential.