2 x (1+ x)(2+ x) d x=a |1+ x|-b |2+ x|+c then the value of a and b is ----

Step-by-step solution

Using 2x = 2 x x : I = 2 x x (1+ x)(2+ x) dx Let t = x , dt = x dx : I = 2t (1+t)(2+t) dt Partial fractions: 2t (1+t)(2+t) = A 1+t + B 2+t t = -1 : -2 = A(1) A = -2 t = -2 : -4 = B(-1) B = 4 I = ( -2 1+t + 4 2+t )dt = -2 |1+t| + 4 |2+t| + c Substituting back t = x : I = -2 |1+ x| + 4 |2+ x| + c Comparing with a |1+ x| - b |2+ x| + c : a = -2 and -b = 4 b = -4