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## Techniques of Integration

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**Techniques of Integration**Dr. Litan Kumar Saha Associate Professor Department of Applied Mathematics University of Dhaka**7.2 INTEGRATION BY PARTS**First, start with the Product Rule for differentiation. This is the formula for Integration by Parts.**Integration by parts**Let dv be the most complicated part of the original integrand that fits a basic integration Rule (including dx). Then uwill be the remaining factors. OR Let u be a portion of the integrand whose derivative is a function simpler than u. Then dvwill be the remaining factors (including dx).**When deciding what to choose for u, remember L I P E T.**L - logarithmic function I - inverse trig function P - polynomial function E - exponential function T - trigonometry function This is usually the preference order in which you would want to choose u.**Integration by parts**u = x dv = exdx du = dx v = ex**Integration by parts**u = lnxdv= x2dx du = 1/x dx v = x3 /3**Integration by parts**u = x2 dv = sin x dx du = 2x dx v = -cos x u = 2xdv = cos x dx du = 2dx v = sin x**Example 4**Notice that when we choose u and dv the second time, u the second time is du from the first, and dv this time is v from the first time.**If we were to use Integration by Parts on the last integral**we would have: u = 2 dv = -cos x dx du = 0 v = -sin x Again u this time is du from the previous integral, and dv this time is v from the previous integral. First of all, finishing this integral we have: Keep this answer handy and let’s take a different look at this problem.**We’ll choose the same u and dv as we did the first time.**But this time, let’s take the derivative of u all the way through and integrate dv for everytime we differentiate u. u dv Notice that when we keep taking the derivative of u it eventually goes to 0, and dv never runs out of integrals. (+) x2 cos x ( - ) 2x sin x (+) 2 -cos x If we multiply each u by its respective v and we switch the signs each time we add each part together we get… 0 -sin x x2 sin x + 2x cos x - 2 sin x + C This method is called tabular integration. This works when the derivatives of u eventually reach 0 and the integrals of dv never end. This is much quicker than using the formula over and over again.