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"If"int(dx)/(x^(3)(1+x^(6))^(23))=xf(x)(...

`"If"int(dx)/(x^(3)(1+x^(6))^(23))=xf(x)(1+x^(6))^(1/3)+C` where, C is a constant of integration, then the function f(x) is equal to

A

` ( 3 ) /( x ^ 3 ) `

B

` - ( 1 ) / ( 2x ^ 3 ) `

C

`- ( 1 ) / ( 2 x ^ 2 ) `

D

` - (1 )/(6x ^ 3 ) `

Text Solution

Verified by Experts

`int (dx ) /(x ^ 3 ( 1 + x^6)^( 2//3)) rArr int (dx ) /(x ^ 7 (1 + (1)/(x^ 6))^ (2//3 )) rArr (1 + (1 ) /(x^ 6)) = t " " - (6 ) /(x ^ 7) dx = dt rArr - (1 ) /(6) int (dt ) /(t ^( 2//3 )) `
` - (1 ) /(6) (( t^( 1//3))/( (1)/(3))) = - (1)/(2) [ ( 1 + ( 1 )/(x^ 6)) ^( 1//3 ) ] + C = - (1)/(2) ((1 + x ^6)^(1//3))/( x ^ 6 )+ C = x f (x) (1 + x ^6 )^( 1//3 ) `
` f (x ) = - ( 1 ) / ( 2x ^ 3) `
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