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# If ∫ (x 2 -1) / [x (x 4 +3 x2 +1) 1/2 ] dx  is m log [ { (x2+1) + (x4+3x2+1) 1/2 } / x ] dx +C then show that m is 1.

12 years ago

A simple method is,just differentiate the solution,

we have

I = ∫ (x2 - 1) / [x (x4+3x2+1)1/2 ] dx  = m log [{(x2+1) + (x4+3x2+1)1/2 } / x ] dx +C

so, on differentiation

m* x / {(x2+1) + (x4+3x2+1)1/2* { 2x2 + [(2x4+3x2)/(x4+3x2+1)1/2 ] - [(x2+1) + (x4+3x2+1)1/2] / x2

m* {1 / x*{(x2+1) + (x4+3x2+1)1/2 }* { (x2 -1).(x4+3x2+1)1/2 + (x4-1 )} / (x4+3x2+1)1/2

m* [{(x2 -1).{(x2+1) + (x4+3x2+1)1/2 } ] / [x*{(x2+1) + (x4+3x2+1)1/2 }*(x4+3x2+1)1/2 ]

m* [{(x2 -1)] / [x*(x4+3x2+1)1/2 ]

which shows m = 1

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