lim_(x→0) (1/x^2 )[∫^(x^2 +(π/3))


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Question Number 78021 by jagoll last updated on 13/Jan/20

$\lim_{x \to 0} \frac{1}{x^{2}} [{\int_{\frac{π}{3}}}^{x^{2} + \frac{π}{3}} \frac{\cos x}{x} d x] =$
Commented by mr W last updated on 13/Jan/20

$\lim_{x \to 0} \frac{1}{x^{2}} [{\int_{\frac{π}{3}}}^{x^{2} + \frac{π}{3}} \frac{\cos x}{x} d x]$ $= \lim_{x \to 0} \frac{\frac{\cos (x^{2} + \frac{π}{3})}{x^{2} + \frac{π}{3}} \times 2 x}{2 x}$ $= \lim_{x \to 0} \frac{\cos (x^{2} + \frac{π}{3})}{x^{2} + \frac{π}{3}}$ $= \frac{3}{π} \times \frac{1}{2} = \frac{3}{2 π}$
Commented by jagoll last updated on 13/Jan/20

$b y u s i n g f u n d a m e n t a l c a l c u l u s t h e o r e m$ $s i r ? t h a n k s$
Commented by mr W last updated on 13/Jan/20

$i {d o n}^{'} t k n o w w h a t i s ‘ ‘ f u n d a m e n t a l$ $c a l c u l u s {t h e o r e m}^{″} .$
Commented by jagoll last updated on 13/Jan/20

$\frac{d}{d x} [\underset{0}{\int^{x}} f (u) d u] = f (x) s i r$
Commented by mr W last updated on 13/Jan/20

$g e n e r a l l y :$ $\frac{d}{d x} [\underset{h (x)}{\int^{g (x)}} f (u) d u] = f (g (x)) g^{'} (x) - f (h (x)) h^{'} (x)$
Commented by jagoll last updated on 13/Jan/20

$o k a y s i r t h a n k y o u$
Commented by msup trace by abdo last updated on 13/Jan/20
$let A(x)=∫_(π/3) ^(x^2 +(π/3)) ((cost)/t)dt ∃ c ∈](π/3),x^2 +(π/3)[ / A(x)=(1/c) ∫_(π/3) ^(x^2 +(π/3)) cost dt =(1/c)(sin(x^2 +(π/3))−((√3)/2)) ⇒((A(x))/x^2 ) =((sin(x^2 +(π/3))−((√3)/2))/(cx^2 )) =(((1/2)sin(x^2 )+((√3)/2)cos(x^2 )−((√3)/2))/(cx^2 )) ∼(1/(2cx^2 )){x^2 +(√3)(1−(x^4 /2))−(√3)} =(1/(2c))−(x^2 /(4c)) ⇒lim_(x→0) A(x) =(1/(2×(π/3))) =(3/(2π)) .$
$l e t A (x) = \int_{\frac{π}{3}}^{x^{2} + \frac{π}{3}} \frac{c o s t}{t} d t$ $\exists c \in] \frac{π}{3}, x^{2} + \frac{π}{3} [/$ $A (x) = \frac{1}{c} \int_{\frac{π}{3}}^{x^{2} + \frac{π}{3}} c o s t d t$ $= \frac{1}{c} (s i n (x^{2} + \frac{π}{3}) - \frac{\sqrt{3}}{2})$ $\Rightarrow \frac{A (x)}{x^{2}} = \frac{s i n (x^{2} + \frac{π}{3}) - \frac{\sqrt{3}}{2}}{{c x}^{2}}$ $= \frac{\frac{1}{2} s i n (x^{2}) + \frac{\sqrt{3}}{2} c o s (x^{2}) - \frac{\sqrt{3}}{2}}{{c x}^{2}}$ $\sim \frac{1}{2 {c x}^{2}} {x^{2} + \sqrt{3} (1 - \frac{x^{4}}{2}) - \sqrt{3}}$ $= \frac{1}{2 c} - \frac{x^{2}}{4 c} \Rightarrow {l i m}_{x \to 0} A (x)$ $= \frac{1}{2 \times \frac{π}{3}} = \frac{3}{2 π} .$
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