int_3^(ab+2) f(x)dx=int_3^(a+2) f(x)dx+int_3^(b+2) f(x)dx where a,b in (1,oo) find f(x)

Charlie Conner

Charlie Conner

Answered question

2022-09-30

f ( x ) is a differentiable function defined for x>2 satisfying
3 a b + 2 f ( x ) d x = 3 a + 2 f ( x ) d x + 3 b + 2 f ( x ) d x
where a , b ( 1 , ) ) and find f(x)

Answer & Explanation

Toby Barron

Toby Barron

Beginner2022-10-01Added 7 answers

f(x) is a differentiable function defined for x>2 satisfying
3 a b + 2 f ( x ) d x = 3 a + 2 f ( x ) d x + 3 b + 2 f ( x ) d x
where a , b ( 1 , ) find f ( x )
Take derivative to a
b f ( a b + 2 ) = f ( a + 2 )
Let a=1
f ( b + 2 ) = f ( 3 ) b
Let x = b + 2
f ( x ) = f ( 3 ) x 2
where f ( 3 ) is some constant c = f ( 3 ), then we have
f ( x ) = c x 2

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