If ₂^ e [ 1 x - 1 ( x)^2 ] dx = a + b 2 , then values of a and b are

Step-by-step solution

Let I = ₂^ e [ 1 x - 1 ( x)^2 ] dx Using integration by parts on the first term: 1 x dx = x x - x ( - 1 ( x)^2 ) 1 x dx = x x + 1 ( x)^2 dx Therefore, [ 1 x - 1 ( x)^2 ] dx = x x Evaluating the definite integral from 2 to e : I = [ x x ]₂^ e = e e - 2 2 = e - 2 2 Comparing this with a + b 2 , we get a = e and b = -2 . Answer: a = e, b = -2