This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ...
View
Full Document
 Spring '06
 Lucas
 Calculus, Integration By Parts

Click to edit the document details