6.1 Completed Notes.pdf - x m Integration by Parts A...

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Unformatted text preview: xfl\‘ , ”m“ \ Integration by Parts ' A. Formula (and how to use it) -';_. -- «tf., Judv=uv"IVdu 4 1 I " 7 I ' I 7' To use the above formula 1. Look at the product that is given, and decide what u will be and What 6171? will be. (We will always put dx in our function with whatever we declare dv to be.) Find du by taking the derivative of u and, Find V by integrating dv Plug into the formula and simplify 9‘95”!" Iv du should be something that you can easily integrate. If not, you may need to switch your choice of u and dV. Example: Ix'Sin(x) dx Wewillletj[x][sm(x)dx] = [[ufidv] So u = x and dv = Sin(x) dx du 3 — :1 2) Divide each side by dx dx i dv . 3 Multiply each side by dx :>_ g- : sm(x) x :> duzdx => v=Isin(x)dx => v z —cos(x) We may now use the formula I” dv = u ‘V-IV du Ix - sin(x) dx x - (— cos (x))—I(—— cos (x)) dx = —x-oos (30+st (x)dx = —x-cos (x)+sin(x)+C Deslré Taylor Math 1242 ' Examples: . (a. MN- , x, K J 1.)y;.‘;7;? '———-*> "M Se 05" <yEZEb 0L“. —; 01» oLflW Desiré Taylor Math 1242 Desiré Taylor Math 1242 Desiré Taylor Math 1242 Desiré Taylor Math 1242 B. The LIATE Principle for Integration by Parts. ) A common question when using integrations by parts is: "What part ofthe equation should I let u be equal to and dV be equal to?” Although there is no "set in stone” answer, we can use the LIATE principle as a guideline __-____._____._=__HWW ° , {adv , '=‘ u {v 412.32,; This rule ofihfli‘ifi) is for choosing the function that is to be u when using integration by parts. Logarithmic functions (Ex: |n(x)) . A verse trigonometric functions (Ex: sin‘ilxll QP igebTaitgms ~ 0 (Ex: x2 or 5x3 +4):2 —x) i ) Trigonometric functions (Ex: cosix” Exponential functions (Ex: i2“r or 829‘} The higher a type offunction appears on this list, the more likely It should serve as u in the integration by parts formula. Conversely, the lower a type of function appears on this list, the more likely it should serve as V. at.) Desiré Taylor Math 1242 Sufi/V: Duh—S \I‘OU’L 15. (Int) Book I’wElicm 43 . f") Suppose lhat f(5) = 2,f(7) = (J,f’(5) =2 9, f’ (7) = 7 and f” \ is continuous. 7 _ Fil1j‘lgtiljgkagg‘o;thpifjl11It‘eintegraIL—tym’aflf? :_ X DlCX) i ; _. S “? 16x) M 5 ==>< \f = 915*) , W ,,~::;:mmwm'm -. 11““. mm); ow =19 “000’“ 535 3“ 5575;?” m» 1; :7-7 ”5'01“" [1361343033] 2‘: LM~L1FS’ C(o’lj $0 5;) Desiré Taylor Math 1242 ...
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