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lec30 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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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
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 =

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lec30 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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