Education LinksLeaving Cert
|
1. The Idea: Integration is the opposite to differentiation. It could be called anti-differentiation. Notation: Think of a function which, when you differentiate it with respect to x, you get f(x). Let u = that part of the integrand such that when you differentiate it you get a multiple of another part of the integrand. 4. Types of Integrals of the LC Course: [A] Alint: Integrals with pure algebraic functions. [B] Expint: Integrals with any hint of the exponential function. [C] Logint: The inverse linear function. [D] Trigint: Integrals with pure trig functions. [E] Specials: Three algebraic functions that require unusual substitutions. [A] Alint The basis of Alint is ![]() ![]() except when p = -1 when ![]() In words: Add one to the power and divide by the new power for all powers except -1 when the answer is ln x. Note 1: ![]() Note 2: ![]() Types of ALINT: Type 1 - Straights: Sums of multiples of ![]() Steps Example 1Evaluate![]() Solution Example 2Evaluate![]() Solution Type 2 - Products and quotients of Algebraic functions which cannot be simplified into sums of multiples of ![]() Steps Example 3Evaluate![]() Solution Example 4Evaluate![]() Solution Example 5Evaluate![]() Solution Example 6Evaluate![]() Solution [B] Expint The basis of Expint is ![]() ![]() Types of EXPINT: Type 1 - Straights: Expofunctions which can be simplified into sums of expofunctions of the form ![]() Steps ![]() Example 7Evaluate![]() Solution Example 8Evaluate![]() Solution Type 2 - Products and Quotients of Expo and non-expo functions. Steps Example 9Evaluate![]() Solution Example 10Evaluate![]() Solution Type 3 - Products and quotients of pure expofunctions which cannot be simplified into a sum of expofunctions of the form ![]() Steps Example 11Evaluate![]() Solution Example 12Evaluate![]() Solution [C] Logint This is the integration of the inverse linear function. It is based on ![]() However, this can be extended to ![]() Example 13Evaluate![]() Solution |
![]() ![]() ![]() |