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Question Number 49922 by Pk1167156@gmail.com last updated on 12/Dec/18

	Answered by tanmay.chaudhury50@gmail.com last updated on 12/Dec/18
	$∫((x^4 dx)/(x^4 (x^4 +(1/x^4 )))) ∫(dx/(x^4 +(1/x^4 ))) ∫(dx/((x^2 +(1/x^2 ))^2 −2)) (1/2)∫(((1−(1/x^2 )+1+(1/x^2 )))/((x^2 +(1/x^2 ))^2 −2)) =(1/2)[∫((d(x+(1/x)))/({(x+(1/x))^2 −2}^2 −2))+∫((d(x−(1/x)))/({(x−(1/x))^2 +2}^2 −2))] =(1/2)[∫(dp/((p^2 −2)^2 −2))+∫(dq/((q^2 +2)^2 −2))] =(1/2)[∫(dp/(p^4 −4p^2 +2))+∫(dq/(q^4 +4q^2 +2))] =(1/2)[∫((1/p^2 )/(p^2 +(2/p^2 )−4))dp+∫((1/q^2 )/(q^2 +(2/q^2 )+4))]=(1/2)(I_1 +I_2 ) now solving I_1 ∫(([(1+((√2)/p^2 ))−(1−((√2)/p^2 ))]×(1/(2(√2))))/((p^2 +(2/p^2 ))−4))dp =(1/(2(√2)))∫((d(p−((√2)/p)))/((p−((√2)/p))^2 +2(√2) −4))−(1/(2(√2)))∫((d(p+((√2)/p)))/((p+((√2)/p))^2 −2(√2) −4)) =(1/(2(√2)))∫((d(p−((√2)/p)))/((p−((√2)/p))^2 −((√(4−2(√2))) )^2 )) −(1/(2(√2)))∫((d(p+((√2)/p)))/((p+((√2)/p))^2 −((√(4+2(√2))) )^2 )) =(1/(2(√2) ))×(1/(2((√(4−2(√2))))))×ln{(((p−((√2)/p))−(√(4−2(√2))))/((p−((√2)/p))+(√(4−2(√2)))))}−(1/(2(√2)))×(1/(2(√(4+2(√2))) ))ln{(((p+((√2)/p))−(√(4+2(√2))))/((p+((√2)/p))+(√(4+2(√2)))))}+c_1 p=(x+(1/x)) pls put p=x+(1/x) nxt going to find I_2 I=(1/2)(I_1 +I_2 ) ←required answer... now findngI_2 ∫((1/q^2 )/((q^2 +(2/q^2 ))+4))dq =(1/(2(√2)))∫(((1+((√2)/q^2 ))−(1−((√2)/q^2 )))/((q^2 +(2/q^2 ))+4))dq =(1/(2(√2)))∫((d(q−((√2)/q)))/((q−((√2)/q))^2 +2(√2) +4))−(1/(2(√2)))∫((d(q+((√2)/q)))/((q+((√2)/q))^2 +4−2(√2))) =(1/(2(√2)))×(1/(√(4+2(√2))))tan^(−1) (((q−((√2)/q))/(√(4+2(√2)))))−(1/(2(√2)))×(1/(√(4−2(√2))))tan^(−1) (((q+((√2)/q))/(√(4−2(√2)))))+c_2 pls put q=(x−(1/x)) so I=(1/2)(I_1 +I_2 )$
	$\int \frac{x^{4} d x}{x^{4} (x^{4} + \frac{1}{x^{4}})}$ $\int \frac{d x}{x^{4} + \frac{1}{x^{4}}}$ $\int \frac{d x}{{(x^{2} + \frac{1}{x^{2}})}^{2} - 2}$ $\frac{1}{2} \int \frac{(1 - \frac{1}{x^{2}} + 1 + \frac{1}{x^{2}})}{{(x^{2} + \frac{1}{x^{2}})}^{2} - 2}$ $= \frac{1}{2} [\int \frac{d (x + \frac{1}{x})}{{{(x + \frac{1}{x})}^{2} - 2}^{2} - 2} + \int \frac{d (x - \frac{1}{x})}{{{(x - \frac{1}{x})}^{2} + 2}^{2} - 2}]$ $= \frac{1}{2} [\int \frac{d p}{{(p^{2} - 2)}^{2} - 2} + \int \frac{d q}{{(q^{2} + 2)}^{2} - 2}]$ $= \frac{1}{2} [\int \frac{d p}{p^{4} - 4 p^{2} + 2} + \int \frac{d q}{q^{4} + 4 q^{2} + 2}]$ $= \frac{1}{2} [\int \frac{\frac{1}{p^{2}}}{p^{2} + \frac{2}{p^{2}} - 4} d p + \int \frac{\frac{1}{q^{2}}}{q^{2} + \frac{2}{q^{2}} + 4}] = \frac{1}{2} (I_{1} + I_{2})$ $n o w s o l v i n g I_{1}$ $\int \frac{[(1 + \frac{\sqrt{2}}{p^{2}}) - (1 - \frac{\sqrt{2}}{p^{2}})] \times \frac{1}{2 \sqrt{2}}}{(p^{2} + \frac{2}{p^{2}}) - 4} d p$ $= \frac{1}{2 \sqrt{2}} \int \frac{d (p - \frac{\sqrt{2}}{p})}{{(p - \frac{\sqrt{2}}{p})}^{2} + 2 \sqrt{2} - 4} - \frac{1}{2 \sqrt{2}} \int \frac{d (p + \frac{\sqrt{2}}{p})}{{(p + \frac{\sqrt{2}}{p})}^{2} - 2 \sqrt{2} - 4}$ $= \frac{1}{2 \sqrt{2}} \int \frac{d (p - \frac{\sqrt{2}}{p})}{{(p - \frac{\sqrt{2}}{p})}^{2} - {(\sqrt{4 - 2 \sqrt{2}})}^{2}} - \frac{1}{2 \sqrt{2}} \int \frac{d (p + \frac{\sqrt{2}}{p})}{{(p + \frac{\sqrt{2}}{p})}^{2} - {(\sqrt{4 + 2 \sqrt{2}})}^{2}}$ $= \frac{1}{2 \sqrt{2}} \times \frac{1}{2 (\sqrt{4 - 2 \sqrt{2}})} \times l n {\frac{(p - \frac{\sqrt{2}}{p}) - \sqrt{4 - 2 \sqrt{2}}}{(p - \frac{\sqrt{2}}{p}) + \sqrt{4 - 2 \sqrt{2}}}} - \frac{1}{2 \sqrt{2}} \times \frac{1}{2 \sqrt{4 + 2 \sqrt{2}}} l n {\frac{(p + \frac{\sqrt{2}}{p}) - \sqrt{4 + 2 \sqrt{2}}}{(p + \frac{\sqrt{2}}{p}) + \sqrt{4 + 2 \sqrt{2}}}} + c_{1}$ $p = (x + \frac{1}{x})$ $p l s p u t p = x + \frac{1}{x}$ $n x t g o i n g t o f i n d I_{2}$ $I = \frac{1}{2} (I_{1} + I_{2}) \leftarrow r e q u i r e d a n s w e r . . .$ $n o w {f i n d n g I}_{2}$ $\int \frac{\frac{1}{q^{2}}}{(q^{2} + \frac{2}{q^{2}}) + 4} d q$ $= \frac{1}{2 \sqrt{2}} \int \frac{(1 + \frac{\sqrt{2}}{q^{2}}) - (1 - \frac{\sqrt{2}}{q^{2}})}{(q^{2} + \frac{2}{q^{2}}) + 4} d q$ $= \frac{1}{2 \sqrt{2}} \int \frac{d (q - \frac{\sqrt{2}}{q})}{{(q - \frac{\sqrt{2}}{q})}^{2} + 2 \sqrt{2} + 4} - \frac{1}{2 \sqrt{2}} \int \frac{d (q + \frac{\sqrt{2}}{q})}{{(q + \frac{\sqrt{2}}{q})}^{2} + 4 - 2 \sqrt{2}}$ $= \frac{1}{2 \sqrt{2}} \times \frac{1}{\sqrt{4 + 2 \sqrt{2}}} {t a n}^{- 1} (\frac{q - \frac{\sqrt{2}}{q}}{\sqrt{4 + 2 \sqrt{2}}}) - \frac{1}{2 \sqrt{2}} \times \frac{1}{\sqrt{4 - 2 \sqrt{2}}} {t a n}^{- 1} (\frac{q + \frac{\sqrt{2}}{q}}{\sqrt{4 - 2 \sqrt{2}}}) + c_{2}$ $p l s p u t q = (x - \frac{1}{x})$ $s o I = \frac{1}{2} (I_{1} + I_{2})$
	Commented by Pk1167156@gmail.com last updated on 12/Dec/18
	thank you very much sir.
	Commented by tanmay.chaudhury50@gmail.com last updated on 12/Dec/18

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