Let f:(11$$,∞$$)→(0$$,∞$$) be given by $$f(x)=$$∏$$l=110$$ $$1(x$$−$$l)where$$ for real number $$a1$$..........an∏$$l=1n$$ al denotes the product $$a1$$×$$a2$$⋯×anStatement $$1$$: ∫$$f(x)dx=$$∑$$l=110$$ $$($$−$$1)llog$$⁡|x−l|(l$$−$$1)!(10$$−$$l)!Statement $$2$$: For x∈[$$11$$,∞$$)f(x)=$$∑$$l=110$$ Alx−l where $$A1=$$∏$$j=110$$ jl−$$jl=1$$,$$2$$,$$10$$

Question

Let f:(11$$,∞$$)→(0$$,∞$$) be given by $$f(x)=$$∏$$l=110$$ $$1(x$$−$$l)where$$ for real number $$a1$$..........an∏$$l=1n$$ al denotes the product $$a1$$×$$a2$$⋯×anStatement $$1$$: ∫$$f(x)dx=$$∑$$l=110$$ $$($$−$$1)llog$$⁡|x−l|(l$$−$$1)!(10$$−$$l)!Statement $$2$$: For x∈[$$11$$,∞$$)f(x)=$$∑$$l=110$$ Alx−l where $$A1=$$∏$$j=110$$ jl−$$jl=1$$,$$2$$,$$10$$

Infinity Learn · Accepted Answer

$$f(x)=$$∏$$l=110$$ $$1x$$−$$l=A1x$$−$$1+A2x$$−$$2+$$⋯$$+A10x$$−$$10where$$ $$Ai=1(i$$−$$1)$$⋯(i$$−$$(i+1))⋯(i$$−$$10)               $$=($$−$$1)i(i$$−$$1)$$!(10$$−$$i)!∫$$f(x)dx=$$∑$$i=110$$ ∫Aix−$$idx=$$∑$$i=110$$ Ailog⁡|x−i|So statement $$1$$ is true but not statement $$2$$.

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Let f:(11,∞)→(0,∞) be given by f(x)=∏l=110 1(x−l)where for real number a1..........an∏l=1n al denotes the product a1×a2⋯×anStatement 1: ∫f(x)dx=∑l=110 (−1)llog⁡|x−l|(l−1)!(10−l)!Statement 2: For x∈[11,∞)f(x)=∑l=110 Alx−l where A1=∏j=110 jl−jl=1,2,10

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