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Integration by Parts: “Undoing” the Product Rule for Derivatives Integrating both sides, we get: ∫udv = ∫d (uv) −∫vdu Or ∫udv =uv−∫vdu ∫vdu should be simpler that the original ∫udv The integral If two functions are not related by derivatives (substitution does not apply), choose one function to be the u (to differentiate) and the other function to be the dv (to integrate)

Integration by Parts: “Undoing” the Product Rule for Derivatives Look at the derivative of a product of functions: d dv du uv = u +v ( ) dx dx dx Let’s use the differential form: d (uv ) =udv+ vdu And solve for udv udv =d (uv) −vdu Integrating both sides, we get: ∫udv = ∫d (uv) −∫vdu

Replay QoE measurement Old way: QoE = Server + Network Modern way: QoE = Servers + Network + Browser Browsers are smart Parallelism on multiple connections JavaScript execution can trigger additional queries Rendering introduces delays in resource access Caching and pre-fetching HTTP replay cannot approximate real Web browser access to resources 0.25s 0.25s 0.06s 1.02s 0.67s 0.90s 1.19s 0.14s 0.97s 1.13s 0.70s 0.28s 0.27s 0.12s 3.86s 1.88s Total network time GET /wiki/page 1 Analyze page GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET combined.min.css jquery-ui.css main-ltr.css commonPrint.css shared.css flaggedrevs.css Common.css wikibits.js jquery.min.js ajax.js mwsuggest.js plugins...js Print.css Vector.css raw&gen=css ClickTracking.js Vector...js js&useskin WikiTable.css CommonsTicker.css flaggedrevs.js Infobox.css Messagebox.css Hoverbox.css Autocount.css toc.css Multilingual.css mediawiki_88x31.png 2 Rendering + JavaScript GET GET GET GET GET GET GET GET GET ExtraTools.js Navigation.js NavigationTabs.js Displaytitle.js RandomBook.js Edittools.js EditToolbar.js BookSearch.js MediaWikiCommon.css 3 Rendering + JavaScript GET GET GET GET GET GET GET GET GET GET GET 4 GET GET GET GET GET GET page-base.png page-fade.png border.png 1.png external-link.png bullet-icon.png user-icon.png tab-break.png tab-current.png tab-normal-fade.png search-fade.png Rendering search-ltr.png arrow-down.png wiki.png portal-break.png portal-break.png arrow-right.png generate page send files send files mBenchLab – [email protected] BROWSERS MATTER FOR QOE? send files send files + 2.21s total rendering time 6

Integration by Parts Back to: ∫udv =uv−∫vdu ∫x ln(x)dx Choose u (to differentiate (“du”)) dv (to integrate (“v”)) u =ln(x) 1 du = dx x dv = xdx x2 v= 2 x2 ⎛ x 2⎞⎞ ⎛ 1 ⎞ ∫x ln(x)dx = 2 ln(x) −∫ ⎜⎝ 22⎟⎠⎟⎠dx⎜⎝ x ⎟⎠ dx x2 x2 = ln(x) − + C 2 4 This second integral is simpler than the first

Integration by Parts Every differentiation rule has a corresponding integration rule. For instance, the Substitution Rule for integration corresponds to the Chain Rule for differentiation. The rule that corresponds to the Product Rule for differentiation is called the rule for integration by parts. The Product Rule states that if f and g are differentiable functions, then [f (x)g (x)] = f (x)g (x) + g (x)f (x) 3

9.8 public, protected and private Inheritance Base class member access specifier Type of inheritance public inheritance protected inheritance private inheritance public in derived class. Can be accessed directly by any non-static member functions, friend functions and nonmember functions. protected in derived class. Can be accessed directly by all non-static member functions and friend functions. private in derived class. Can be accessed directly by all non-static member functions and friend functions. protected in derived class. Can be accessed directly by all Protected non-static member functions and friend functions. protected in derived class. Can be accessed directly by all non-static member functions and friend functions. private in derived class. Can be accessed directly by all non-static member functions and friend functions. Hidden in derived class. Can be accessed by non-static member functions and friend functions through public or protected member functions of the base class. Hidden in derived class. Can be accessed by non-static member functions and friend functions through public or protected member functions of the base class. Public Private Hidden in derived class. Can be accessed by non-static member functions and friend functions through public or protected member functions of the base class. 2003 Prentice Hall, Inc. All rights reserved. 83

11 9.3 Inheritances Base class member access specifier Type of inheritance public inheritance protected inheritance private inheritance public in derived class. Can be accessed directly by any non-static member functions, friend functions and nonmember functions. protected in derived class. Can be accessed directly by all non-static member functions and friend functions. private in derived class. Can be accessed directly by all non-static member functions and friend functions. protected in derived class. Can be accessed directly by all Protected non-static member functions and friend functions. protected in derived class. Can be accessed directly by all non-static member functions and friend functions. private in derived class. Can be accessed directly by all non-static member functions and friend functions. Hidden in derived class. Can be accessed by non-static member functions and friend functions through public or protected member functions of the base class. Hidden in derived class. Can be accessed by non-static member functions and friend functions through public or protected member functions of the base class. Public Private Hidden in derived class. Can be accessed by non-static member functions and friend functions through public or protected member functions of the base class. 2003 Prentice Hall, Inc. All rights reserved.

Definite Integrals Using Substitution Two Methods 1. Integrate the function by substitution, and write the answer back in terms of the original variable. Then evaluate the limits of integration. 2. Integrate the function by substitution and also write the limits of integration in terms of substitution variable (normally u), evaluating the integral in terms of the substitution variable.

Integration by Parts: “Undoing” the Product Rule for Derivatives ∫x ln(x)dx Consider: We have no formula for this integral. Notice that x and ln(x) are not related by derivatives, so we cannot use the substitution method.