Question

Let f(x) be a function satisfying f ′(x) = f(x) with f(0) = 1 and g be the function satisfying f(x) + g(x) = x. Then the value of ∫f(x)g(x)dx for x ∈ [0, 1] is

(A) (3 - e²)/2

(B) (e² - 3)/2

(C) e²/2

(D) (e - 2)/4

Answer 1

Answer is (A) (3 - e²)/2

If′(x) = f(x) 

Integrating, we get 

log f(x) = x + k ⇒ f(x) = e^x+k 

Now, f(0) = 1 

Therefore, 

 1 = e⁰⋅ e^k 

⇒ k = 0 

Therefore, f(x) = e^x 

and g(x) = x – f(x) = x – e^x 

Therefore,