Evaluate $$\int\limits_{-1}^{\frac 3 2}|x\sin(\pi x)| dx $$
Can you help me find the domains in which the function will be positive and in which it be negative? I mean how to determine in which range it will positive or negative?
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1Hint: Can you compute the zeros of this function? – Stefan Mar 17 '15 at 10:07
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another hint: calculate sign of $x$ and $\sin (\pi x)$ separately at first. – mathreadler Mar 17 '15 at 10:09
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Yes thanks I got positive for -1 to 1 and negative for 1 to 3/2. – Maini who Mar 17 '15 at 10:12
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Without the absolute value bars, the integrand is an even function. Then for nonnegative $x$, $\sin \pi x$ is nonnegative on intervals $[2n,2n+1]$ for $n=0,1,2,\ldots$, so the same is true of $x\sin\pi x$.
By evenness, then, you can extend the result to the intervals $[-(2n+1),-2n]$ for $n=0,1,2,\ldots$ as well.
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