Let f ( x ) be a function satisfying f ′ ( x ) = f ( x ) with f ( 0 ) = 1 and g ( x ) be a function that satisfies f ( x ) + g ( x ) = x 2 . Then the value of the integral ∫ 0 1 f ( x ) g ( x ) d x

Question

Let f ( x ) be a function satisfying f ′ ( x ) = f ( x ) with f ( 0 ) = 1 and g ( x ) be a function that satisfies f ( x ) + g ( x ) = x 2 . Then the value of the integral ∫ 0 1 ​ f ( x ) g ( x ) d x , is

TestHub · Accepted Answer

Let f ( x ) = e x ∴ ∫ 0 1 ​ f ( x ) g ( x ) d x = ∫ 0 1 ​ e x ( x 2 − e x ) d x = ∫ 0 1 ​ x 2 e x d x − ∫ 0 1 ​ e 2 x d x = [ x 2 e x ] 0 1 ​ − 2 [ xe x − e x ] 0 1 ​ − 2 1 ​ [ e 2 x ] = e − [ 2 e 2 ​ − 2 1 ​ ] − 2 [ e − e + 1 ] = e − 2 e 2 ​ − 2 3 ​

Mathematics - Definite Integration Question with Solution | TestHub

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