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Theorem Triangle Inequality for IntegrationProof
Example Show that
Integration by Parts Theorem Integration by Parts Let or Example Show that (a) AL84II-1(a) For any non-negative integer k, let Express Hence, or otherwise, evaluate (b) For any non-negative integers (i) Show that if
(ii) Evaluate (iii) Show that if
(iv) Evaluate AL94II-11 For any non-negative integer (a) (i) Show that [ Note: You may assume without proof that (ii) Using (i), or otherwise, evaluate (iii) Show that (b) For (i) Using (a)(iii), or otherwise, express (ii) Evaluate
Example Solution Let
Example Show that (a) (b) Deduce that
Continuity and Differentiability of a Definite Integral Theorem Mean Value Theorem for Integral If Proof
Theorem Continuity of definite Integral If Proof
Theorem * Fundamental Theorem of Calculus Let Proof
Remark : Proof
Example Evaluate the Derivatives of the following (i) (ii) (iii) Solution
ExampleLet
Example Let Prove that
AL90II-5(a) Evaluate (b) If
ExampleEvaluate
AL97II-5(b) Evaluate
AL98II-2 Let (a) Evaluate (b) Using (a), or otherwise, show that
ExampleLet Evaluate (a) (b) (c)
Example Suppose
Remark
Improper Integrals
Definition A definite integral
Example
Definition (a) (b) (c) ( Or (d) If
for any
Definition The improper integral is said to be Convergent or Divergent according to the improper integral exists or not. Example Evaluate (a)
Example Evaluate Example Evaluate
Theorem Let
Example Discuss whether
Example For any non-negative integer
(a) Show if Hence, if (b) Find
Date: 2016-04-22; view: 1044
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