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Integrating over piecewise functions 
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#1
Feb1513, 05:12 PM

P: 346

given a function [itex]f(x)[/itex] that is piecewise smooth on interval [itex]L<x<L[/itex] except at [itex]N1[/itex] points, is [itex]\int_{L}^L f'(x)dx [/itex] legal or would i have to [tex]\sum_{i=1}^N \int_{x_i}^{x_{i+1}} f'(x)dx[/tex]
where [itex]x_{N+1}=L[/itex] and [itex]x_{1}=L[/itex] also, am i correct that if [itex]f(x)[/itex] is piecewise smooth, then [itex]f'(x)[/itex] is piecewise continuous but not necessarily piecewise smooth? thanks in advance! 


#2
Feb1513, 06:27 PM

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$$f'(x) = \begin{cases} \frac{1}{2x^{1/2}} & \textrm{ if }x > 0 \\ \frac{1}{2x^{1/2}} & \textrm{ if }x < 0 \\ \end{cases}$$ As ##f'## is unbounded on ##[L,L]##, it's necessary to use two (improper) integrals to integrate it: $$\lim_{a \rightarrow 0^} \int_{L}^{a} f'(x) dx + \lim_{b \rightarrow 0^+}\int_{b}^{L} f'(x) dx$$ Both limits exist and the answers have opposite signs, so the result is 0. 


#3
Feb1713, 02:40 PM

P: 346

thanks for the reply. think i have it now.



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