# Chapter 3(B) - Fundamental Theorem of Calculus(Part II(II...

This preview shows pages 1–10. Sign up to view the full content.

Fundamental Theorem of Calculus (Part II) (II) If is an of on [, ], then F f ab antiderivative ( ) [ ( )] ( ) () b b a a fx d x Fx F b Fa = =-

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Example [ ] 0 ) 0 cos 2 (cos cos sin 2 0 2 0 = - - = - = p p p x dx x . sin 2 0 dx x p 0 x p 2p y -1 1 y = sin x Evaluate
Example [ ] 1 1 1 ln (l n ln1) 1 e e d xx x e = =- = . 1 1 dx x e 0 x 1 e y x y 1 = Evaluate

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Various Integration Techniques
Integration by Substitution - Example C x x C u du u dx x x x + - + = + = = + - + 3 2 3 2 2 2 ) 3 2 ( 6 1 6 1 2 1 ) 1 ( ) 3 2 ( 22 Evaluate ( 2 3 ) ( 1) . x x x dx + -+ 2 Let 2 3. u xx =+- Then 2 ( 1). du x dx =+ 2 ( 1) d u x dx 1 2 ( d ux dx

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Integration by Substitution - Example C x C u du u dx x x + = + = = 5 5 4 4 sin 5 1 5 1 cos sin 4 Evaluate si n co s . x x dx Let si n. ux = co s d u = Then co s. du x dx =
Integration by Substitution - Example C x C u du u dx x x + = + = = 6 6 5 5 ) (ln 6 1 6 1 ) (ln 5 (l n) Evaluate . x dx x Let l n. ux = 1 Then . du d xx = 1 d u dx x =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Integration by Substitution - Example xx x x e xe u u e e d x e e dx e du eC + = = =+ ∫∫ Evaluate . x e dx + Note that . x e e ee + = Let . x ue = m n mn e + = x d u e dx = Then . x du e dx =
Integration by Substitution - Example 4 2 4 2 0 0 ta n1 ta n se c 22 x x x dx p p  == 

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 11/01/2011 for the course MATH 1505 taught by Professor Yap during the Spring '11 term at National University of Singapore.

### Page1 / 22

Chapter 3(B) - Fundamental Theorem of Calculus(Part II(II...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online