The value of ` int ( ( x ^ 2 + 1 )) /( x ^ 4 + x ^ 2 + 1 ) dx ` is

Options

A. ` (1 ) /( sqrt 3 ) tan ^( - 1 ) { ( x - 1// x ) /( sqrt 3)} + C `
B. ` ( 1 ) /( 2sqrt 3 ) log { ( ( x- 1 // x) - sqrt 3) /( (x - 1 // x ) + sqrt 3 } + C `
C. ` tan ^( - 1 ) (( x + 1// x ) /(sqrt 3) ) + C `
D. `tan^( - 1 ) (( x - 1//x )/ (sqrt 3 )) + C `

Answer

A

Explanation

Let ` I= int ( x^ 2 + 1 ) /( x ^4 + x^ 2 + 1 ) dx `
` = int (( 1 + ( 1) / ( x^ 2 )))/ ( ( x ^ 2 + 1 + ( 1 ) / ( x ^ 2 )) ) dx `
` = int (( 1 + ( 1 )/ ( x ^ 2 ))) /( ( x - ( 1 )/( x) ) ^ 2 + (sqrt 3 )^ 2 ) dx `
` = int (dt )/( ( sqrt 3 )^ 2 + t^ 2 ) `
[ let ` t = x - ( 1 ) / ( x ) rArr dt = ( 1 + ( 1 )/ ( x ^ 2 ) ) dx] `
` = ( 1 )/(sqrt 3 ) tan ^( - 1 ) (( t )/ (sqrt 3 ) ) + C `
` = ( 1 )/ ( sqrt 3 ) tan^( - 1 ) { ( 1 )/ (sqrt3 ) ( x - ( 1 )/ (x )) } + C `