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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 ‘VIV du
Ix  sin(x) dx x  (— cos (x))—I(—— cos (x)) dx
= —xoos (30+st (x)dx = —xcos (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; :77 ”5'01“" [1361343033]
2‘: LM~L1FS’ C(o’lj $0 5;) Desiré Taylor Math 1242 ...
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 Fall '06
 Lucas
 Calculus, Integration By Parts

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