Let f be a differentiable function satisfying f (x) = 2 + x2 – \int_{0}^{x} t \textrm{ } f \left(\right. t \left.\right) \textrm{ } dt \textrm{ }\textrm{ } \forall x \in R then List-I List-II (P) f (e) is equal to (1) 0 (Q) f ' (e) is equal to (2) 1 (R) area bounded by f (x) and its image in y = x line and the co-ordinates axes is equal to (3) 2 (S) \int_{0}^{2} f \left(\right. x \left.\right) \textrm{ } dx is equal to (4) 4

Engineering

Mathematics

Leibnitz Rule Derivative of Anti Derivatives

Question

Let f be a differentiable function satisfying f (x) = 2 + x² – $\int_{0}^{x} t f (t) dt \forall x \in R$ then

List-I	List-II
(P) f (e) is equal to	(1) 0
(Q) f ' (e) is equal to	(2) 1
(R) area bounded by f (x) and its image in y = x line and the co-ordinates axes is equal to	(3) 2
(S) $\int_{0}^{2} f (x) dx$ is equal to	(4) 4

P → 3

Q → 1

R → 4

S → 2

P → 2

Q → 1

R → 2

S → 3

P → 3

Q → 1

R → 4

S → 4

P → 3

Q → 3

R → 4

S → 1

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f ' (x) = 2x – x f (x)

f ' (x) + x f (x) = 2x

Solving above differential equation, we get

f (x) = 2

Now, verify the options.