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Unformatted text preview: MATH 118, LECTURE 2: Integration by Parts 1 Integration By Parts Consider trying to take the integral of integraldisplay xe x dx. We could try to find a substitution but would quickly grow frustrated — there is no substitution we can make which would simplify the integral. Instead we can rely on information we know for the product rule for differ- entiation. If we let u and v be functions of x , we know from the product rule for differentiation that d dx [ uv ] = du dx v + u dv dx . We want to use this to tell us something about integration, not differentia- tion, but what can we do? Well, let’s just integrate over x and see what we get. Since integration undoes differentiation, we obtain uv = integraldisplay du dx v dx + integraldisplay u dv dx dx. We can now simplify the differentials ( dx ) and rearrange to get integraldisplay u dv = uv- integraldisplay v du. (1) This is the Integration by Parts formula and it is one of the primary tools for integration. It can also be stated for definite integrals as integraldisplay b a u dv = [ uv ] b a- integraldisplay b a v du. (2) It is probably not immediately obvious how this helps us, since we still have to evaluate the integral on the right-hand side. The trick is that, if we make appropriate choices for the parts of the original integral — u and dv — it may turn out upon differentiating u and integrating dv to get du and 1 v , respectively, that the integral on the right-hand side is easier to evaluate than the one the left. So, ideally, we replace the original integral with a simpler one! Let’s see if the above integral can be solved in this matter. Our first task is to select u and dv from the expression xe x dx . One of these quantities will be differentiated, and one integrated, in order to form our new quantity to be integrated. It turns out the appropriate choice for this problem is u = x dv = e x dx du = dx v = e x . Now using formula (1) we have integraldisplay xe x dx = xe x- integraldisplay e x dx....
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This note was uploaded on 04/14/2009 for the course ENGINEERIN MATH 118 taught by Professor Soares during the Spring '09 term at Waterloo.
- Spring '09