integrals proving equality

Question

Prove that $$\int_a^b f(x)\,dx = c\int_{a/c}^{b/c} f(cy)\,dy$$

I've tried to use subtitution rule, but I'm afraid I can't do this if f is not continuous.

Thanks.

you can use substitution. It is legal here. – Guy Mar 15 '14 at 14:36 — Guy, Mar 15 '14 at 14:36
can you explain please why it's legal here? – Shirly Geffen Mar 15 '14 at 14:39 — Shirly Geffen, Mar 15 '14 at 14:39
The substitution $y=x/c$ is differentiable. – Unwisdom Mar 15 '14 at 14:41 — Unwisdom, Mar 15 '14 at 14:41

score 4 · Answer 1 · answered Mar 15 '14 at 14:41

4

Substituting $x=g\left(y\right)$ is allowed here under certain conditions on function $g$ (so not conditions on the integrand $f$). Here you do that with function $g\left(y\right)=cy$ which is okay if $c\neq0$.

answered Mar 15 '14 at 14:41

drhab

151,093

This really answers the question. The asker forgot to mention that $c \neq 0$, and you brought this up. Yes. This is certainly true. – NasuSama Mar 15 '14 at 14:45

score 0 · Answer 2 · answered Mar 15 '14 at 14:42

0

Let $x=cy$. Therefore, $dx=c dy$, and the limits become the desired ones: $a/c$ and $b/c$. Therefore, the integral is:

$$\int_a^b f(x)\,dx = c\int_{a/c}^{b/c} f(cy)\,dy$$

Which is what we wanted.

answered Mar 15 '14 at 14:42

Imanol Pérez Arribas

647
4
16

does not answer the question. – Guy Mar 15 '14 at 14:43
@Sabyasachi. Why ? This is correctly proven as long as $c$ is not zero. – Claude Leibovici Mar 15 '14 at 14:50
@ClaudeLeibovici the OP did consider the substitution. (S)he isn't sure if it is correct. That explanation is required. – Guy Mar 15 '14 at 15:03
The OP's confusion regarding the requirement on the continuity of $f$(which is wrong) should also have been cleared up. – Guy Mar 15 '14 at 15:04

integrals proving equality

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