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

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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 = =-
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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
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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
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Various Integration Techniques
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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
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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 =
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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 =
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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 =
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Integration by Substitution - Example 4 2 4 2 0 0 ta n1 ta n se c 22 x x x dx p p  == 
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Chapter 3(B) - Fundamental Theorem of Calculus (Part II)...

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