Let $f(x)$ be a function satisfying $f(x)=f(\frac{100}{x})$ $\forall x>0$. If $\int _{1} ^{10} \frac{f(x)}{x}dx=5$, then find the value of $\int _{1} ^{100} \frac{f(x)}{x}dx$.
Basically we need $\int _{10} ^{100} \frac{f(x)}{x}dx$. In $\int _{1} ^{10} \frac{f(x)}{x}dx$, I replaced $t=10x$ to get limits from $10$ to $100$ but I am stuck with $f(t/10)$. Could someone help me with this?
We know that $\int_1^{10}\frac{f(x)}{x} dx = 5$
Now in that integral change variables to $t=\frac{100}{x}$. We find
$\int_1^{10}\frac{f(x)}{x} dx = \int_{10}^{100}\frac{f(\frac{100}{t})}{t}dt$
But of course the value of the integral didn't change since we just substituted. Using $f(\frac{100}{t})=f(t)$, we thus have
$\int_{10}^{100}\frac{f(t)}{t}dt=5$
Finally, we find
$\int_1^{100}\frac{f(x)}{x}dx=\int_1^{10}\frac{f(x)}{x}dx+\int_{10}^{100}\frac{f(x)}{x}dx=5+5=10$