Question

Integration

Evaluate :

$\int \limits_ { - \frac{1}{ \sqrt{2} } }^{ \frac{1}{ \sqrt{2} } } \frac{ {x}^{8} }{1 - {x}^{4} } \times [ { \sin}^{ - 1} (1 - 2 {x}^{2} + { \cos}^{ - 1} (2x \sqrt{1 - {x}^{2} } ]dx$

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Answer 1

Question :

Integrate

$\int \limits_ { - \frac{1}{ \sqrt{2} } }^{ \frac{1}{ \sqrt{2} } } \frac{ {x}^{8} }{1 - {x}^{4} } \times [ { \sin}^{ - 1} (1 - 2 {x}^{2}) + { \cos}^{ - 1} (2x \sqrt{1 - {x}^{2} }) ]dx$

Formula's used :

1)$sin{}^{-1}x+cos{}^{-1}x=\frac{\pi}{2}$ , x∈ [ -1,1]

2)$tan{}^{-1}x+cot{}^{-1}x=\frac{\pi}{2}$ , x∈R

3)$sec{}^{-1}x+cosec{}^{-1}x=\frac{\pi}{2}$ , x ∈(-∞,-1)⋃ (1,∞)

$4)2 \sin {}^{ - 1} (x) = \sin {}^{ - 1} (2x \sqrt{1 - x {}^{2} } )$

$5)2 \cos {}^{ - 1} (x) = \cos {}^{ - 1} (2x{}^{2}-1 )$

Solution :

$\int \limits_ { - \frac{1}{ \sqrt{2} } }^{ \frac{1}{ \sqrt{2} } } \frac{ {x}^{8} }{1 - {x}^{4} } \times [ { \sin}^{ - 1} (1 - 2 {x}^{2}) + { \cos}^{ - 1} (2x \sqrt{1 - {x}^{2} } )]dx$

$=\sf\:\pi [\frac{1}{2}tan{}^{-1} (\frac{1}{\sqrt{2}})+\dfrac{1}{4}\log(\dfrac{\sqrt{2}+1}{\sqrt{2}-1}-\dfrac{21}{20\sqrt{2}}]$

Refer to the attachment.

Answer 2

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Refer to the attachment

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