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A study Guide on Integration by Parts Produced by Bryce Borchers and Keldon Reller Revised by Learning Center Integration by Parts is a method used when the integrand is a product of functions. This method is used because the integral of a product of functions is not the product of the individual integrals of those functions. The method of integration by parts transforms the integral into a form much easier to integrate. Integration by parts takes the form ó f (x ) $g' (x ) dx = f (x ) $g(x ) K ó f ' (x )$g(x ) dx õ õ This equation is also known and more commonly referred to as ó u $ dv = u$v K ó v $ du õ õ When breaking down an integral into parts the first step is to decide which function will be (u) and which function will be (dv). There are a lot of different ways to determine which function is which, but we will teach a specific method, the LIATE method. The LIATE method determines which function should be (u). Steps of the LIATE method are as follows. 1) First write out LIATE L – Logarithmic I – Inverse Trig A – Algebraic T – Trig E – Exponential 2) Whichever function corresponds with the highest term on the list gets assigned to the all important (u) 3) Lastly, the other function gets defined as (dv). Once the functions have been defined as either (u) or (dv) a table can and should be created to find (du) and (v), needed to complete the equation of integration by parts. A popular table commonly used looks like this. u = f(x) du = f’(x)dx dv = g’(x)dx v = g(x) The next step is to set up the equation ó u $ dv = u$v K ó v $ du õ õ and solve it. Now we will run through some example problems to assess your understanding of integration by parts. Sample 1) Solve the integral F ( x ) x cos xdx Solution Notes F ( x ) x cos xdx ux du dx dv cos xdx v sin xdx F ( x ) x sin x sin xdx F ( x ) x sin x cos x C Sample 2) Solve the integral F ( x ) x 2e x dx Solution Notes F ( x ) x e dx 2 x u x2 du 2xdx dv e x dx v e x dx F ( x ) x 2e x 2xe x dx u 2x du 2dx dv e x dx v e x dx F ( x ) x 2e x 2xe x 2e x dx F ( x ) x 2e x 2xe x 2e x C 3 Sample 3) Solve the integral F ( x ) xe x dx 1 Solution Notes 3 F ( x ) xe x dx 1 F ( x ) xe x 3 1 3 e x dx 1 3 F ( x ) 3e 3 e 1 e x dx 1 F ( x ) 3e 3 e 1 e 3 e 1 2e 3 2e 1