Transcription of 25Integration by Parts
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25Integration by Parts
Page 1 of 7 3 Formula =vduuvudv I. Guidelines for Selecting u and dv: (There are always exceptions, but these are generally helpful.) L-I-A-T-E Choose u to be the function that comes first in this list: L: Logrithmic Function I: Inverse Trig Function A: Algebraic Function T: Trig Function E: Exponential Function Example A: dxxxln3 *Since lnx is a logarithmic function and 3xis an algebraic function, let: u = lnx (L comes before A in LIATE) dv = 3xdx du = x1 dx v = =443xdxx =vduuvxdxxln3 dxxxxx144)(ln44 = dxxxx =34414)(ln Cxxx+ =441)(ln444 Cxxx+ =16)(ln444 ANSWER Integration By Parts Page 2 of 7 Example B: dxxx)ln(cossin u = ln(cosx) (Logarithmic Function) dv = sinx dx (Trig Function [L comes before T in LIATE]) du = dxxdxxxtan)sin(cos1 = v = =xdxxcossin =vduuvdxxx)ln(cossin =dxxxxx)tan)(cos()cos))((ln(cos =dxxxxxxcossin)(cos)ln(coscos =dxxxxsin)ln(coscos Cxxx++ =cos)ln(coscos ANSWER Example C: dxx1sin *At first it appears that integration by Parts does not apply, but let: xu1sin = (Inverse Trig Function) dxdv1= (Algebraic Function) dxxdu211 = ==xdxv1 =
Example C: ∫sin−1 x dx *At first it appears that integration by parts does not apply, but let: u =sin−1 x (Inverse Trig Function) dv =1 dx (Algebraic Function) dx x du 1 2 1 − = v =∫1dx = x ∫∫sin−1 x dx = uv − vdu dx x x x ∫x − = − − 2 1 1 1 (sin )( ) x x ⎟∫ −x − x dx ⎠ ⎞ ⎜ ⎝ ⎛ = − − − (1 )− ...
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