Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is the integral $$\int_1^\infty\frac{x^{-a} - x^{-b}}{\log(x)}\,dx$$ convergent, where $b>a>1$?

I think the answer lies in defining a double integral with $yx^{(-y-1)}$ and applying Tonelli's Theorem, but the integral of $\frac{x^{-a}}{\log x}$ is still not integrable. Any ideas?

share|improve this question
add comment

1 Answer

up vote 2 down vote accepted

If we make substitution $x=\log t$, then we get integral of the form $$ \int\limits_{0}^\infty\frac{e^{-(a-1)t}-e^{-(b-1)t}}{t}dt $$ Now result follows from Frullani's integral formula applied to the function $f(t) = e^{-t}$. Moreover, this formula gives us exact value of the integral $$ \int\limits_{0}^\infty\frac{e^{-(a-1)t}-e^{-(b-1)t}}{t}dt=(f(0)-f(\infty))\log\frac{b-1}{a-1}=\log\frac{b-1}{a-1} $$

As for the proof of this formula see this discussion.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.