Question

Prove that the function is a constant on the interval (0, INFINITY)
F(x)= integral(2x,x) 1/t dt
Pls tell me what I have to do here to prove

Answer 1

the derivative of a constant is what?

Answer 2

zero

Answer 3

then if the derivative if F(x) = 0, then F(x) must be a constant

Answer 4

its pretty simple to integrate 1/t; but there is a pretty neat rule to avoid that route

Answer 5

\[\Large F(x)=\int_{a(x)}^{b(x)}f(t)~dt\]
\[\Large F(x)=F[b(x)]-F[a(x)]\]
\[\Large \frac{d}{dx}F(x)=\frac{d}{dx}F[b(x)]-\frac{d}{dx}F[a(x)]\]
\[\Large \frac{d}{dx}F(x)=f[b(x)]~b'-f[a(x)]~a'\]

Answer 6

might have to do some limits for improper integrals to finish the proof for the interval x=0 to inf

Answer 7

ok i'll try

Answer 8

good luck ;)