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lecture34hout - MATH 006 Calculus and Linear...

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MATH 006 Calculus and Linear Algebra (Lecture 34) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 12
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Outline 1 Integration by Parts Albert Ku (HKUST) MATH 006 2 / 12
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Integration by Parts Integration by Parts The following is the integration by parts formula : Z udv = uv - Z vdu How to use the above formula? Consider the following example: Example Find Z xe x dx . Note: method of substitution will not work for this indefinite integral. Albert Ku (HKUST) MATH 006 3 / 12
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Integration by Parts Solution To use the integration by parts formula, we need to split “ xe x dx ” into two parts: the “ u ” part and the “ dv ” part. Notice that the “ dv ” must include dx ”. In this example, we let u = x , dv = e x dx . Then we need to compute “ du ” and “ v ”: du = dx , v = Z e x dx = e x . According to the integration by parts formula, we have Z xe x dx = xe x - Z e x dx . Z xe x dx = xe x - e x + C Albert Ku (HKUST) MATH 006 4 / 12
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Integration by Parts Example Find Z x ln xdx .
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