Integrate x*ln(x)

This example provides us with a better understanding of how "Integration by parts" works. Building an extensive knowledge of Integral calculus requires a lot of work, but once the basics are fully understood, it follows naturally that one would go into a more deep approach.

Recall: ∫fg′=fg−∫f′g (Integration by parts), where f and g are well-defined functions

Approach:

1) Think before you act

2) Think two steps ahead

3) Be clear and not verbose

Solution:

 ∫xlnx dx=

= ∫ [(x^2)/2]' lnx dx=

= [(x^2)/2]lnx - ∫[(x^2)/2] (lnx)' dx=

= [(x^2)/2]lnx - 1/2∫(x^2)*(1/x) dx=

= [(x^2)/2]lnx - 1/2∫xdx=

= [(x^2)/2]lnx - 1/2 * (x^2)/2   +C (don't forget the constant)=

= (simple manipulation) [(x^2)(2lnx -1)]/4 +C

Now remember, Maths can be very easy once you take action. The best car in the world will not take you to the right place if you don't know where you want to go. #naturalenthusiasm

Answered by Maximilian M. Maths tutor

5456 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Find the Binomial Expansion of (1-5x)^4.


The line y=5-x intersects the curve y=x^2-3x+2 at the points P and Q. Find the (x,y) coordinates of P and Q.


Using Integration by Parts, find the indefinite integral of ln(x), and hence show that the integral of ln(x) between 2 and 4 is ln(a) - b where a and b are to be found


How do you integrate ln(x) ?


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences