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The graph of the antiderivative of f x = x e x 2 passes through 0 , 3 , then the value of g 2 - f 0 is

Correct answer

7

Step-by-step solution

Let g x represent the antiderivative of f x Therefore, g x = ∫ x e x 2 d x ‍ = x 2 e x 2 - ∫ 1 ⋅ 2 e x 2 d x ‍ …[integration by parts] = 2 x e x 2 - 4 e x 2 + c The graph of y = g ( x ) passes through ( 0 , 3 ) ⇒ 3 = 0 - 4 e 0 + c ⇒ c = 7 Hence, g ( x ) = 2 x e x 2 - 4 e x 2 + 7 ⇒ g ( x ) = 2 ( x - 2 ) e x 2 + 7 ⇒ g ( 2 ) = 7 , f ( 0 ) = 0 ⇒ g ( 2 ) - f ( 0 ) = 7

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