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lec30 - Lecture 30 18.01 Fall 2006 Lecture 30 Integration...

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Lecture 30 18.01 Fall 2006 Lecture 30: Integration by Parts, Reduction Formulae Integration by Parts Remember the product rule: ( uv ) = u v + uv We can rewrite that as uv = ( uv ) - u v Integrate this to get the formula for integration by parts: uv dx = uv - u v dx Example 1. tan - 1 x dx . At first, it’s not clear how integration by parts helps. Write tan - 1 x dx = tan - 1 x (1 dx ) = uv dx · with u = tan - 1 x and v = 1 . Therefore, 1 v = x and u = 1 + x 2 Plug all of these into the formula for integration by parts to get: 1 tan - 1 x dx = uv dx = (tan - 1 x ) x - 1 + x 2 ( x ) dx = x tan - 1 x - 1 2 ln | 1 + x 2 | + c Alternative Approach to Integration by Parts As above, the product rule: ( uv ) = u v + uv can be rewritten as uv = ( uv ) - u v This time, let’s take the definite integral: b b b uv dx = ( uv ) dx - u v dx a a a 1
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Lecture 30 18.01 Fall 2006 By the fundamental theorem of calculus, we can say b b b uv dx = uv u v dx a - a a Another notation in the indefinite case is u dv = uv - v du This is the same because dv = v dx = uv dx = u dv and du = u dx = u v dx = vu dx = v du Example 2. (ln x ) dx 1 u = ln x ; du = dx and dv = dx ; v = x x 1 (ln x ) dx = x ln x - x dx = x ln x - dx = x ln x - x + c x We can also use “advanced guessing” to solve this problem.
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