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Question Number 63273 by mathmax by abdo last updated on 01/Jul/19
let F(x) =∫_x^2 ^x^3 ((sin(t))/(t+x)) dt 1) calculate lim_(x→0) F(x) and lim_(x→+∞) F(x) 2)calculste lim_(x→0) F^′ (x) and lim_(x→+∞) F^′ (x)
letF(x)=∫x2x3sin(t)t+xdt1)calculatelimx→0F(x)andlimx→+∞F(x)2)calculstelimx→0F′(x)andlimx→+∞F′(x)
Commented by mathmax by abdo last updated on 03/Jul/19
1) ∃c ∈]x^2 ,x^3 [ wich verify ∫_x^2 ^x^3 ((sint)/(t+x))dt =sinc ∫_x^2 ^x^3 (dt/(t+x)) =sinc [ln∣t+x∣]_(t=x^2 ) ^(t=x^3 ) =sinc {ln∣x^3 +x∣−ln∣x^2 +x∣} =lnc ln∣((x^2 +1)/(x+1))∣ ⇒lim_(x→0) F(x) =lim_(c→0) ×lim_(x→0) ln∣((x^2 +1)/(x+1))∣ =0 we have x^2 ≤t ≤x^3 ⇒x^2 +x ≤t+x≤x+x^3 ⇒for x>0 (1/(x+x^3 ))≤(1/(t+x)) ≤(1/(x^2 +x)) ⇒ (1/(x+x^3 )) ≤∣((sint)/(t+x))∣≤(1/(x^2 +x)) but ∣F(x)∣≤∫_x^2 ^x^3 ∣((sint)/(t+x))∣dt →0 (x→+∞) because lim_(x→+∞) (1/(x+x^3 )) =lim_(x→+∞) (1/(x^2 +x)) =0
1)∃c∈]x2,x3[wichverify∫x2x3sintt+xdt=sinc∫x2x3dtt+x=sinc[ln∣t+x∣]t=x2t=x3=sinc{ln∣x3+x∣−ln∣x2+x∣}=lncln∣x2+1x+1∣⇒limx→0F(x)=limc→0×limx→0ln∣x2+1x+1∣=0wehavex2⩽t⩽x3⇒x2+x⩽t+x⩽x+x3⇒forx>01x+x3⩽1t+x⩽1x2+x⇒1x+x3⩽∣sintt+x∣⩽1x2+xbut∣F(x)∣⩽∫x2x3∣sintt+x∣dt→0(x→+∞)becauselimx→+∞1x+x3=limx→+∞1x2+x=0

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