find approcimstive value of ∫_(π/3) ^(π/2) (x/(sinx))dx


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Question Number 88414 by abdomathmax last updated on 10/Apr/20

$f i n d a p p r o c i m s t i v e v a l u e o f \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{x}{s i n x} d x$
Commented by mathmax by abdo last updated on 12/Apr/20
$we have sinx =x−(x^3 /6) +....⇒x−(x^3 /6)≤sinx≤x ∀ x∈[(π/3),(π/2)] ⇒ (1/x)≤(1/(sinx))≤(1/(x−(x^3 /6))) ⇒ 1≤(x/(sinx))≤(1/(1−(x^2 /6))) ⇒ (π/2)−(π/3)≤∫_(π/3) ^(π/2) (x/(sinx))dx ≤ 6∫_(π/3) ^(π/2) (dx/(6−x^2 )) we have ∫_(π/3) ^(π/2) (dx/(6−x^2 )) =(1/(2(√6)))∫_(π/3) ^(π/2) ((1/((√6)−x))+(1/((√6)+x)))dx =(1/(2(√6)))[ln∣(((√6)+x)/((√6)−x))∣]_(π/3) ^(π/2) =(1/(2(√6))){ ln((((√6)+(π/2))/((√6)−(π/2))))−ln((((√6)+(π/3))/((√6)−(π/3))))} =(1/(2(√6))){ln(((2(√6)+π)/(2(√6)−π)))−ln(((3(√6)+π)/(3(√6)−π)))} ⇒ (π/6)≤∫_(π/3) ^(π/2) (x/(sinx))dx ≤((√6)/2){ ln(((2(√6)+π)/(2(√6)−π)))−ln(((3(√6)+π)/(3(√6)−π)))} we can take v_0 =(π/2) +((√6)/4){ ln(((2(√6)+π)/(2(√6)−π)))−ln(((3(√6)+π)/(3(√6)−π)))} as a approximste value$
$w e h a v e s i n x = x - \frac{x^{3}}{6} + . . . . \Rightarrow x - \frac{x^{3}}{6} ⩽ s i n x ⩽ x \forall x \in [\frac{π}{3}, \frac{π}{2}] \Rightarrow$ $\frac{1}{x} ⩽ \frac{1}{s i n x} ⩽ \frac{1}{x - \frac{x^{3}}{6}} \Rightarrow 1 ⩽ \frac{x}{s i n x} ⩽ \frac{1}{1 - \frac{x^{2}}{6}} \Rightarrow$ $\frac{π}{2} - \frac{π}{3} ⩽ \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{x}{s i n x} d x ⩽ 6 \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{d x}{6 - x^{2}}$ $w e h a v e \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{d x}{6 - x^{2}} = \frac{1}{2 \sqrt{6}} \int_{\frac{π}{3}}^{\frac{π}{2}} (\frac{1}{\sqrt{6} - x} + \frac{1}{\sqrt{6} + x}) d x$ $= \frac{1}{2 \sqrt{6}} {[l n ∣ \frac{\sqrt{6} + x}{\sqrt{6} - x} ∣]}_{\frac{π}{3}}^{\frac{π}{2}} = \frac{1}{2 \sqrt{6}} {l n (\frac{\sqrt{6} + \frac{π}{2}}{\sqrt{6} - \frac{π}{2}}) - l n (\frac{\sqrt{6} + \frac{π}{3}}{\sqrt{6} - \frac{π}{3}})}$ $= \frac{1}{2 \sqrt{6}} {l n (\frac{2 \sqrt{6} + π}{2 \sqrt{6} - π}) - l n (\frac{3 \sqrt{6} + π}{3 \sqrt{6} - π})} \Rightarrow$ $\frac{π}{6} ⩽ \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{x}{s i n x} d x ⩽ \frac{\sqrt{6}}{2} {l n (\frac{2 \sqrt{6} + π}{2 \sqrt{6} - π}) - l n (\frac{3 \sqrt{6} + π}{3 \sqrt{6} - π})} w e c a n t a k e$ $v_{0} = \frac{π}{2} + \frac{\sqrt{6}}{4} {l n (\frac{2 \sqrt{6} + π}{2 \sqrt{6} - π}) - l n (\frac{3 \sqrt{6} + π}{3 \sqrt{6} - π})} a s a a p p r o x i m s t e v a l u e$
Commented by mathmax by abdo last updated on 12/Apr/20
$v_0 =(π/(12)) +((√6)/4){.....}$
$v_{0} = \frac{π}{12} + \frac{\sqrt{6}}{4} {. . . . .}$
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