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Question Number 54701 by Meritguide1234 last updated on 09/Feb/19

	Answered by behi83417@gmail.com last updated on 09/Feb/19

	${t g}^{- 1} \frac{x}{x^{2} - 1} = a$ ${s i n}^{2} a = \frac{1}{1 + {c o t}^{2} a} = \frac{1}{1 + x^{2} + x^{- 2} - 2} = \frac{x^{2}}{x^{4} - x^{2} + 1}$ $\Rightarrow \sqrt{x^{4} - x^{2} + 1} = \frac{x}{s i n a}$ ${c o s}^{2} a = \frac{1}{1 + {t g}^{2} a} = \frac{1}{1 + \frac{x^{2}}{{(x^{2} - 1)}^{2}}} = \frac{{(x^{2} - 1)}^{2}}{x^{4} - x^{2} + 1}$ $\Rightarrow c o s a = \frac{x^{2} - 1}{\sqrt{x^{4} - x^{2} + 1}}$ $s i n (x + a) = s i n x . \frac{x^{2} - 1}{\sqrt{x^{4} - x^{2} + 1}} + c o s x . \frac{x}{\sqrt{x^{4} - x^{2} + 1}}$ ${t g}^{- 1} \frac{x^{2} - 1}{2 x} = b \Rightarrow {c o s}^{2} b = \frac{1}{1 + \frac{{(x^{2} - 1)}^{2}}{4 x^{2}}} \Rightarrow$ $c o s b = \frac{2 x}{x^{2} + 1}, s i n b = \frac{x^{2} - 1}{x^{2} + 1}$ $s i n (2 x - b) = s i n 2 x . \frac{2 x}{x^{2} + 1} - c o s 2 x . \frac{x^{2} - 1}{x^{2} + 1}$ $\Rightarrow I = \int \frac{(x^{2} - 1) . s i n x + x . c o s x}{[(x^{2} + 1) + (x^{2} - 1) c o s 2 x - 2 x s i n 2 x]} d x$
	Commented by Meritguide1234 last updated on 09/Feb/19

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	Answered by tanmay.chaudhury50@gmail.com last updated on 10/Feb/19
	$tan^(−1) ((x/(x^2 −1))) sec^2 a=((x^4 −2x^2 +1+x^2 )/((x^2 −1)^2 ))=((x^4 −x^2 +1)/((x^2 −1)^2 )) cos^2 a=(((x^2 −1)^2 )/(x^4 −x^2 +1)) cosa=((x^2 −1)/(√(x^4 −x^2 +1))) sin^2 a=1−(((x^4 −2x^2 +1))/(x^4 −x^2 +1))=((x^4 −x^2 +1−x^4 +2x^2 −1)/(x^4 −x^2 +1)) sina=(x/(√(x^4 −x^2 +1))) so sin(x+tan^(−1) (x/(x^2 −1)))=sinx×((x^2 −1)/(√(x^4 −x^2 +1)))+cosx×(x/(√(x^4 −x^2 +1))) so N_r ={(x^2 −1)sinx+xcosx}×(1/(√(x^4 −x^2 +1)))×(√(x^4 −x^2 +1)) N_r =(x^2 −1)sinx+xcosx ★tanb=((x^2 −1)/(2x)) sec^2 b=1+((x^4 −2x^2 +1)/(4x^2 ))=(((x^2 +1)^2 )/((2x)^2 )) cosb=((2x)/((x^2 +1))) sin^2 b=(((x^2 +1)^2 −4x^2 )/((x^2 +1)^2 ))=(((x^2 −1)/(x^2 +1)))^2 sinb=((x^2 −1)/(x^2 +1)) or ((1−x^2 )/(1+x^2 )) sin(2x−tan^(−1) ((x^2 −1)/(2x))) =sin2x×((2x)/(x^2 +1))−co2x×((x^2 −1)/(x^2 +1)) or sin2x×((2x)/(x^2 +1))−cos2x×((1−x^2 )/(1+x^2 )) ((2xsin2x−(x^2 −1)cos2x)/(x^2 +1)) =(((2xsin2x−(x^2 −1)cos2x)/(x^2 +1)) ((2xsin2x−(1−x^2 )cos2x)/(x^2 +1)) so D_r =(x^2 +1){1−((2xsin2x−(x^2 −1)cos2x)/(x^2 +1))} =x^2 +1−2xsin2x+(x^2 −1)cos2x =x^2 +1−4xsinxcosx+x^2 cos2x−cos2x =x^2 (1+cos2x)+1−cos2x−4xsinxcosx =2x^2 cos^2 x+2sin^2 x−4xsinxcosx =2(xcosx−sinx)^2 or2(sinx−xcosx)^2 or D_r =(x^2 +1){1−((2xsin2x−(1−x^2 )cos2x)/(x^2 +1))} =x^2 +1−2xsin2x+(1−x^2 )cos2x =x^2 +1−4xsinxcosx+cos2x−x^2 cos2x =x^2 (1−cos2x)+1+cos2x−4xsinxcosx =2x^2 sin^2 x+2cos^2 x−4xsinxcosx =2(xsinx−cosx)^2 or2(cosx−xsinx)^2 I_(blue) =∫(((x^2 −1)sinx+xcosx)/(2(xcosx−sinx)^2 ))dx.....1 or ∫(((x^2 −1)sinx+xcosx)/(2(sinx−xcosx)^2 ))dx.......2 I_(red) =∫(((x^2 −1)sinx+xcosx)/(2(xsinx−cosx)^2 ))dx.....3 or∫(((x^2 −1)sinx+xcosx)/(2(cosx−xsinx)^2 ))dx....4 four intregation... (d/dx)(xcosx−sinx)=−xsinx+cosx−cosx=−xsinx (d/dx)(sinx−xcosx)=cosx−(x×−sinx+cosx)=xsinx (1/2)∫(/) (d/(dx ))(xsinx−cosx)=xcosx+sinx+sinx=xcosx+2sinx (d/dx)(cosx−xsinx)=−sinx−xcosx−sinx=−(xcosx+2sinx)$
	${t a n}^{- 1} (\frac{x}{x^{2} - 1})$ ${s e c}^{2} a = \frac{x^{4} - 2 x^{2} + 1 + x^{2}}{{(x^{2} - 1)}^{2}} = \frac{x^{4} - x^{2} + 1}{{(x^{2} - 1)}^{2}}$ ${c o s}^{2} a = \frac{{(x^{2} - 1)}^{2}}{x^{4} - x^{2} + 1} c o s a = \frac{x^{2} - 1}{\sqrt{x^{4} - x^{2} + 1}}$ ${s i n}^{2} a = 1 - \frac{(x^{4} - 2 x^{2} + 1)}{x^{4} - x^{2} + 1} = \frac{x^{4} - x^{2} + 1 - x^{4} + 2 x^{2} - 1}{x^{4} - x^{2} + 1}$ $s i n a = \frac{x}{\sqrt{x^{4} - x^{2} + 1}}$ $s o s i n (x + {t a n}^{- 1} \frac{x}{x^{2} - 1}) = s i n x \times \frac{x^{2} - 1}{\sqrt{x^{4} - x^{2} + 1}} + c o s x \times \frac{x}{\sqrt{x^{4} - x^{2} + 1}}$ $s o N_{r} = {(x^{2} - 1) s i n x + x c o s x} \times \frac{1}{\sqrt{x^{4} - x^{2} + 1}} \times \sqrt{x^{4} - x^{2} + 1}$ $N_{r} = (x^{2} - 1) s i n x + x c o s x$ $★ t a n b = \frac{x^{2} - 1}{2 x} {s e c}^{2} b = 1 + \frac{x^{4} - 2 x^{2} + 1}{4 x^{2}} = \frac{{(x^{2} + 1)}^{2}}{{(2 x)}^{2}}$ $c o s b = \frac{2 x}{(x^{2} + 1)} {s i n}^{2} b = \frac{{(x^{2} + 1)}^{2} - 4 x^{2}}{{(x^{2} + 1)}^{2}} = {(\frac{x^{2} - 1}{x^{2} + 1})}^{2}$ $s i n b = \frac{x^{2} - 1}{x^{2} + 1} o r \frac{1 - x^{2}}{1 + x^{2}}$ $s i n (2 x - {t a n}^{- 1} \frac{x^{2} - 1}{2 x})$ $= s i n 2 x \times \frac{2 x}{x^{2} + 1} - c o 2 x \times \frac{x^{2} - 1}{x^{2} + 1}$ $o r s i n 2 x \times \frac{2 x}{x^{2} + 1} - c o s 2 x \times \frac{1 - x^{2}}{1 + x^{2}}$ $\frac{2 x s i n 2 x - (x^{2} - 1) c o s 2 x}{x^{2} + 1}$ $= \frac{(2 x s i n 2 x - (x^{2} - 1) c o s 2 x}{x^{2} + 1}$ $\frac{2 x s i n 2 x - (1 - x^{2}) c o s 2 x}{x^{2} + 1}$ $s o D_{r} = (x^{2} + 1) {1 - \frac{2 x s i n 2 x - (x^{2} - 1) c o s 2 x}{x^{2} + 1}}$ $= x^{2} + 1 - 2 x s i n 2 x + (x^{2} - 1) c o s 2 x$ $= x^{2} + 1 - 4 x s i n x c o s x + x^{2} c o s 2 x - c o s 2 x$ $= x^{2} (1 + c o s 2 x) + 1 - c o s 2 x - 4 x s i n x c o s x$ $= 2 x^{2} {c o s}^{2} x + 2 {s i n}^{2} x - 4 x s i n x c o s x$ $= 2 {(x c o s x - s i n x)}^{2} o r 2 {(s i n x - x c o s x)}^{2}$ $o r D_{r} = (x^{2} + 1) {1 - \frac{2 x s i n 2 x - (1 - x^{2}) c o s 2 x}{x^{2} + 1}}$ $= x^{2} + 1 - 2 x s i n 2 x + (1 - x^{2}) c o s 2 x$ $= x^{2} + 1 - 4 x s i n x c o s x + c o s 2 x - x^{2} c o s 2 x$ $= x^{2} (1 - c o s 2 x) + 1 + c o s 2 x - 4 x s i n x c o s x$ $= 2 x^{2} {s i n}^{2} x + 2 {c o s}^{2} x - 4 x s i n x c o s x$ $= 2 {(x s i n x - c o s x)}^{2} o r 2 {(c o s x - x s i n x)}^{2}$ $I_{b l u e} = \int \frac{(x^{2} - 1) s i n x + x c o s x}{2 {(x c o s x - s i n x)}^{2}} d x . . . . . 1$ $o r \int \frac{(x^{2} - 1) s i n x + x c o s x}{2 {(s i n x - x c o s x)}^{2}} d x . . . . . . . 2$ $I_{r e d} = \int \frac{(x^{2} - 1) s i n x + x c o s x}{2 {(x s i n x - c o s x)}^{2}} d x . . . . . 3$ $o r \int \frac{(x^{2} - 1) s i n x + x c o s x}{2 {(c o s x - x s i n x)}^{2}} d x . . . . 4$ $f o u r i n t r e g a t i o n . . .$ $\frac{d}{d x} (x c o s x - s i n x) = - x s i n x + c o s x - c o s x = - x s i n x$ $\frac{d}{d x} (s i n x - x c o s x) = c o s x - (x \times - s i n x + c o s x) = x s i n x$ $\frac{1}{2} \int \frac{}{}$ $\frac{d}{d x} (x s i n x - c o s x) = x c o s x + s i n x + s i n x = x c o s x + 2 s i n x$ $\frac{d}{d x} (c o s x - x s i n x) = - s i n x - x c o s x - s i n x = - (x c o s x + 2 s i n x)$
	Commented by tanmay.chaudhury50@gmail.com last updated on 09/Feb/19

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