Let J(x) = e^x ( x-1 x^2+1 )^2 dx . If J(0) = 4 , then the value of 5 J(2) - e^2 is equal to

Step-by-step solution

J(x) = e^x x^2 - 2x + 1 (x^2+1)^2 dx = e^x ( x^2+1 (x^2+1)^2 - 2x (x^2+1)^2 ) dx = e^x ( 1 x^2+1 - 2x (x^2+1)^2 ) dx This is of the form e^x (f(x) + f'(x)) dx = e^x f(x) + C , where f(x) = 1 x^2+1 . Thus, J(x) = e^x x^2+1 + C . Given J(0) = 4 : e^0 0+1 + C = 4 1 + C = 4 C = 3 . So, J(x) = e^x x^2+1 + 3 . At x = 2 : J(2) = e^2 5 + 3 . Therefore, 5 J(2) - e^2 = 5 ( e^2 5 + 3 ) - e^2 = e^2 + 15 - e^2 = 15 . Answer: 15