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I am studying for n exam and don't have solutions for this problem:

Let $f:[0,3] \to R $, so that

$\int_0^3f(x)dx$ exists.

Find a function g and a constant C so that

$\int_0^3f(x)dx= C*\int_a^bg(x)dx$.

g shall depend only on f and C on a and b.

I think I have to solve this by substituting, but am not sure. Thanks for any hint!

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    It isn't immediately visible to me why $g (x) $ can't be $f (x)/C $. – 2017-02-23
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    @Rohan: The limits of integration are different. – 2017-02-23

2 Answers 2

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To rephrase Fred to match the wording of your problem: let (constant) $ g(x) = \int_0^3{f(t)\,dt} $ and $ C = \frac{1}{b-a} $. Then $$ C\int_a^b{g(x)\,dx} = \frac{1}{b-a}\cdot(b-a)\cdot\int_0^3{f(t)\,dt}=\int_0^3{f(t)\,dt} $$

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Hint:

Introduce a new variable $u$ such that $x=\frac{3}{b-a}\cdot u + \frac{3a}{a-b}$