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

Question

If &int; (x 2 -1) / [x (x 4 +3 x 2 +1) 1/2 ] dx is m log [ { (x 2 +1) + (x 4 +3x 2 +1) 1/2 } / x ] dx +C then show that m is 1.

Ramesh V · Answer

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

If ∫ (x ² -1) / [x (x⁴ +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.

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

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.

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If ∫ (x ² -1) / [x (x⁴ +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.