My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Thanks Tree1Thanks
  • 1 Post By skipjack
Reply
 
LinkBack Thread Tools Display Modes
October 7th, 2019, 02:05 AM   #1
Senior Member
 
Joined: Dec 2015
From: somewhere

Posts: 734
Thanks: 98

Cannot solve integral

Evaluate: $\displaystyle \int_{0}^{1} \frac{x^{b}-x^{a}}{\ln(x)}dx\;$ .
$\displaystyle a\neq b$, $\displaystyle a,b>0.$
idontknow is offline  
 
October 7th, 2019, 04:21 AM   #2
Senior Member
 
Joined: Dec 2015
From: somewhere

Posts: 734
Thanks: 98

Quote:
Originally Posted by idontknow View Post
Evaluate: $\displaystyle \int_{0}^{1} \frac{x^{b}-x^{a}}{\ln(x)}dx\;$ .
$\displaystyle a\neq b$, $\displaystyle a,b>0.$
The integral is similar to $\displaystyle I(z) =\int_{0}^{1} \frac{x^z }{\ln x} dx=\frac{1}{z+1} \int_{-\infty}^{0} \frac{e^{t(z+1)}}{t(z+1)}dt=-\frac{1}{z+1 }\int_{0}^{-\infty} (tz+t)^{-1}\cdot \sum_{n=0}^{\infty}\frac{(tz+t)^n }{n!}dt=-\frac{1}{z+1 }\int_{0}^{-\infty} \sum_{n=0}^{\infty}\frac{(tz+t)^{n-1} }{n!}dt$. How can I continue from here ?

Last edited by skipjack; October 7th, 2019 at 04:34 AM.
idontknow is offline  
October 7th, 2019, 04:32 AM   #3
Global Moderator
 
Joined: Dec 2006

Posts: 21,035
Thanks: 2271

It seems to be $\displaystyle \ln\left(\!\frac{b+1}{a+1}\!\right)$.
Thanks from idontknow
skipjack is offline  
October 7th, 2019, 06:11 AM   #4
Senior Member
 
Joined: Dec 2015
From: somewhere

Posts: 734
Thanks: 98

A fast method is : $\displaystyle \displaystyle \int_{0}^{1} \frac{x^{b}-x^{a}}{\ln(x)}dx=\int_{a}^{b}[\int_{0}^{1}x^{\lambda }dx]d\lambda =\int_{a}^{b} [(1+\lambda )^{-1}x^{1+\lambda }|_{0}^{1} ]d\lambda =\int_{a}^{b} \frac{d\lambda }{1+\lambda }=\ln|\frac{b+1}{a+1}|$.
idontknow is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
integral, solve



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
solve integral amirstark Complex Analysis 1 November 23rd, 2015 04:27 PM
how to solve this integral? Rosa Calculus 4 September 27th, 2015 05:31 PM
How can I solve such an Integral? ghafarimahsa Calculus 1 September 11th, 2013 12:07 PM
How to solve an integral. Ad van der ven Calculus 3 December 17th, 2011 09:53 AM
How would you solve this integral? Niko Bellic Calculus 3 December 1st, 2008 07:47 AM





Copyright © 2019 My Math Forum. All rights reserved.