lim-x-0-1-x-2-x-2-pi-3-pi-3-cos-x-x-dx-

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

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} .