aligned & x d x [15] (1+x^2 )¹² (2+x^2 )¹⁸ = ( 1+x^2 2+x^2 )^ 1 / n +C & n = aligned

Step-by-step solution

aligned I & = x d x [15] (1+x^2 )¹² (2+x^2 )¹⁸ & = x d x (1+x^2 )^ 12 / 15 (2+x^2 )^ 18 / 15 & = x d x (1+x^2 )^ 4 / 5 (2+x^2 )^ 6 / 5 & = x d x (1+x^2 )^2 ( 2+x^2 1+x^2 )^ 6 / 5 aligned Put u= 2+x^2 1+x^2 aligned & d u= (1+x^2 )(2 x)- (2+x^2 )(2 x) (1+x^2 )^2 d x & d u= 2 x (1+x^2-2-x^2 ) d x (1+x^2 )^2 & d u= -2 x d x (1+x^2 )^2 & x d x (1+x^2 )^2 =- d u 2 aligned Then, I=- 1 2 d x (u)^ 6 / 5 aligned & =- 1 2 u^ -6 5 +1 (- 6 5 +1 ) +C & = 5 2 (u)^ -1 / 5 +C & = 5 2 ( 1+x^2 2+x^2 )^ 1 / 5 +C aligned Thus, = 5 2 an