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Question Number 41806 by Raj Singh last updated on 13/Aug/18

Commented by prof Abdo imad last updated on 13/Aug/18
$let A = ∫ ((arcsin(.(√x))−arccos((√x)))/(arcsin(√x) +arccos(√x))) dx arcosx +arcsinx)^′ =−(1/(√(1−x^2 ))) +(1/(√(1−x^2 ))) for −1<x<1 ⇒arcoss+arcsinx =c c=(π/2) ⇒A = ∫ ((arcsin(√x)−((π/2)−arcsin(√x)))/(arcsin(√x)+(π/2) −arcsin(√x)))dx =(2/π) ∫ (2arcsin(√x) −(π/2))dx =(4/π) ∫ arcsin((√x))dx −x +c but ∫ arcsin((√x))dx =_((√x)=t) ∫ 2 arcsin(t)t dt =2 ∫ t arcsin(t)dt (by parts) =2{ (t^2 /2) arcsint −∫ (t^2 /(2(√(1−t^2 ))))dt} =t^2 arcsint − ∫ (t^2 /(√(1−t^2 ))) dt but ∫ (t^2 /(√(1−t^2 ))) dt =_(t=sinκ) ∫ ((sin^2 α)/(cosα)) cosα dα =∫ ((1−cos(2α))/2) dα =(1/2) α −(1/4)sin(2α) =(1/2)α −(1/2) cosαsinα =(1/2) arcsint−(1/2)t(√(1−t^2 )) =(1/2) arcsin((√x))−(1/2)(√x)(√(1−x)) ⇒ A =(4/π){x arcsin((√x))−(1/2)arcsin((√x))+(1/2)(√x)(√(1−x))}−x +c A =(4/π){(x−(1/2))arcsin((√x)) +(((√x)(√(1−x)))/2)}−x +c .$
$l e t A = \int \frac{a r c s i n (. \sqrt{x}) - a r c c o s (\sqrt{x})}{a r c s i n \sqrt{x} + a r c c o s \sqrt{x}} d x$ ${a r c o s x + a r c s i n x)}^{^{'}} = - \frac{1}{\sqrt{1 - x^{2}}} + \frac{1}{\sqrt{1 - x^{2}}} f o r$ $- 1 < x < 1 \Rightarrow a r c o s s + a r c s i n x = c$ $c = \frac{π}{2} \Rightarrow A = \int \frac{a r c s i n \sqrt{x} - (\frac{π}{2} - a r c s i n \sqrt{x})}{a r c s i n \sqrt{x} + \frac{π}{2} - a r c s i n \sqrt{x}} d x$ $= \frac{2}{π} \int (2 a r c s i n \sqrt{x} - \frac{π}{2}) d x$ $= \frac{4}{π} \int a r c s i n (\sqrt{x}) d x - x + c b u t$ $\int a r c s i n (\sqrt{x}) d x =_{\sqrt{x} = t} \int 2 a r c s i n (t) t d t$ $= 2 \int t a r c s i n (t) d t (b y p a r t s)$ $= 2 {\frac{t^{2}}{2} a r c s i n t - \int \frac{t^{2}}{2 \sqrt{1 - t^{2}}} d t}$ $= t^{2} a r c s i n t - \int \frac{t^{2}}{\sqrt{1 - t^{2}}} d t b u t$ $\int \frac{t^{2}}{\sqrt{1 - t^{2}}} d t =_{t = s i n κ} \int \frac{{s i n}^{2} α}{c o s α} c o s α d α$ $= \int \frac{1 - c o s (2 α)}{2} d α = \frac{1}{2} α - \frac{1}{4} s i n (2 α)$ $= \frac{1}{2} α - \frac{1}{2} c o s α s i n α = \frac{1}{2} a r c s i n t - \frac{1}{2} t \sqrt{1 - t^{2}}$ $= \frac{1}{2} a r c s i n (\sqrt{x}) - \frac{1}{2} \sqrt{x} \sqrt{1 - x} \Rightarrow$ $A = \frac{4}{π} {x a r c s i n (\sqrt{x}) - \frac{1}{2} a r c s i n (\sqrt{x}) + \frac{1}{2} \sqrt{x} \sqrt{1 - x}} - x + c$ $A = \frac{4}{π} {(x - \frac{1}{2}) a r c s i n (\sqrt{x}) + \frac{\sqrt{x} \sqrt{1 - x}}{2}} - x + c .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 13/Aug/18
	$x=sin^2 ∝ dx=2sin∝cos∝d∝ sin^(−1) (sin∝)+cos^(−1) (sin∝) sin^(−1) (sin∝)+cos^(−1) {cos((Π/2)−∝)} =(Π/2) sin^(−1) (sin∝)−cos^(−1) (sin∝) =2∝−(Π/2) ∫(((2∝−(Π/2))sin2∝.d∝)/(Π/2)) (2/Π)[∫2∝sin2∝d∝−(Π/2)∫sin2∝ d∝] (4/Π)∫∝sin2∝d∝−∫sin2∝d∝ (4/Π)[∝.(((−cos2∝)/2))−∫1.(((−cos2∝)/2))d∝]+((cos2∝)/2)+c (4/Π)[∝.(((−cos2∝)/2))+((sin2∝)/4)]+((cos2∝)/2)+cc cos2∝.(((−2∝)/Π)+(1/2))+sin2∝.(1/Π)+c (((Π−4∝)/(2Π)))cos2∝+(1/Π)sin2∝+c (((Π−4∝)/(2Π)))(cos^2 ∝−sin^2 ∝)+(1/Π)(2sin∝cos∝)+c (((Π−4∝)/(2Π)))(1−x−x)+(1/Π)(2.(√x) .(√(1−x)) )+c (((Π−4sin^(−1) (√x))/(2Π)))(1−2x)+(1/Π)(2(√x) .(√(1−x)) )+c pls check$
	$x = {s i n}^{2} \propto d x = 2 s i n \propto c o s \propto d \propto$ ${s i n}^{- 1} (s i n \propto) + {c o s}^{- 1} (s i n \propto)$ ${s i n}^{- 1} (s i n \propto) + {c o s}^{- 1} {c o s (\frac{Π}{2} - \propto)}$ $= \frac{Π}{2}$ ${s i n}^{- 1} (s i n \propto) - {c o s}^{- 1} (s i n \propto)$ $= 2 \propto - \frac{Π}{2}$ $\int \frac{(2 \propto - \frac{Π}{2}) s i n 2 \propto . d \propto}{\frac{Π}{2}}$ $\frac{2}{Π} [\int 2 \propto s i n 2 \propto d \propto - \frac{Π}{2} \int s i n 2 \propto d \propto]$ $\frac{4}{Π} \int \propto s i n 2 \propto d \propto - \int s i n 2 \propto d \propto$ $\frac{4}{Π} [\propto . (\frac{- c o s 2 \propto}{2}) - \int 1 . (\frac{- c o s 2 \propto}{2}) d \propto] + \frac{c o s 2 \propto}{2} + c$ $\frac{4}{Π} [\propto . (\frac{- c o s 2 \propto}{2}) + \frac{s i n 2 \propto}{4}] + \frac{c o s 2 \propto}{2} + c c$ $c o s 2 \propto . (\frac{- 2 \propto}{Π} + \frac{1}{2}) + s i n 2 \propto . \frac{1}{Π} + c$ $(\frac{Π - 4 \propto}{2 Π}) c o s 2 \propto + \frac{1}{Π} s i n 2 \propto + c$ $(\frac{Π - 4 \propto}{2 Π}) ({c o s}^{2} \propto - {s i n}^{2} \propto) + \frac{1}{Π} (2 s i n \propto c o s \propto) + c$ $(\frac{Π - 4 \propto}{2 Π}) (1 - x - x) + \frac{1}{Π} (2 . \sqrt{x} . \sqrt{1 - x}) + c$ $(\frac{Π - 4 {s i n}^{- 1} \sqrt{x}}{2 Π}) (1 - 2 x) + \frac{1}{Π} (2 \sqrt{x} . \sqrt{1 - x}) + c p l s c h e c k$
	Commented by tanmay.chaudhury50@gmail.com last updated on 13/Aug/18

	$i o v e r l o o k e d t h e \propto . . . l e t m e r e c t i f y . . .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 13/Aug/18

	$r e c t i f i e d b y r e d c o l o u r . . .$
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