Question

The value of `int (dx)/(x(x^(n)+1))`
A. ` ( 1 ) /( n ) log (( x^ n ) /( x ^n + 1 )) + C `
B. `log (( x ^n + 1 ) /( x^n )) + C `
C. ` ( 1 )/( n ) log (( x^n + 1 ) /( x^n )) + C `
D. `log (( x^n ) /( x^n + 1 )) + C `

Answer 1

Correct Answer - A
Let I ` = int ( dx ) / ( x ( x^n + 1 )) `
(let ` t = x^n + 1, dt = nx ^ ( n - 1 ) dx` )
` = int ( dx ) /( nx^n * t ) " " ( ( dt ) /( nx ^n ) = ( dx ) / ( x )) `
` = ( 1 )/ ( n ) int (dt ) /( t ( t - 1 )) `
` = ( 1 )/( n ) int { ( 1 )/( t - 1 ) - ( 1 )/ (t )} dt `
` = ( 1 )/ (n ) { log ( t- 1 ) - log t } + C `
`= ( 1 )/ ( n ) log (t - 1 ) /( t ) +C `
`( 1 )/( n ) log ( (x^n)/(x^n + 1 ) + C `