If $$f(a,b)=\int_a^b \frac {e^{\frac {x^3}{a^3}}- e^{\frac {b^3}{x^3}}}{x} dx$$, $b\neq 0$ then prove that $f(a,b)$ is always non negative.
My try:
$$f(a,b)=\int_a^b \frac {e^{\frac {x^3}{a^3}}- e^{\frac {b^3}{x^3}}}{x} dx= \int_a^b \frac {e^{\frac {x^3}{a^3}}}{x} dx+\int_a^b \frac {-e^{\frac {b^3}{x^3}}}{x} dx$$
$$=\int_a^b \frac {e^{\frac {x^3}{a^3}}}{x^3} x^2 dx +\int_a^b \frac {-x^3\cdot e^{\frac {b^3}{x^3}}}{x^4} dx$$
Now in the first integral let $\frac {x^3}{a^3}=t$ and in the second integral let $\frac {b^3}{x^3}=m$
Using this I get $$f(a,b)=\frac 13 \int_1^{\frac {b^3}{a^3}} \frac {e^t}{t} dt +\frac 13 \int_{\frac {b^3}{a^3}}^1 \frac {e^m}{m} dm$$
This simply gives me $f(a,b)=0$ thus proving the statement but I don't get an intuitive feeling over how can this function be 0. So I think I am going wrong. Can someone please point out what I am missing in this question?