Consider x ⁻¹ x d x= A (x^2+1 ) ⁻¹ x+ B x+ C , where C is the constant of integration. What is the value of B?

Step-by-step solution

aligned & x ⁻¹ x d x= A (x^2+1 ) ⁻¹ x+ B x+ C & Let I = x ⁻¹ x d x &= ⁻¹ x ( x^2 2 )- 1 1+x^2 x^2 2 d x &= x^2 2 ⁻¹ x- 1 2 x^2 1+x^2 d x &= x^2 2 ⁻¹ x- 1 2 (1+x^2 )-1 1+x^2 d x &= x^2 2 ⁻¹ x- 1 2 [ 1 d x- 1 1+x^2 d x ] &= x^2 2 ⁻¹ x- 1 2 [x- ⁻¹ x ]+ C &= x^2 2 ⁻¹ x- 1 2 x+ 1 2 ⁻¹ x+ C &= ( x^2+1 2 ) ⁻¹ x- x 2 + C aligned Value of B =- 1 2