Find the integral of x log x

by RoRi

November 3, 2015

Evaluate the following integral:

$\int x \log x \, dx.$

We use integration by parts, defining

$\begin{align*} u &= \log x & du &= \frac{1}{x} \, dx \\ dv &= x \, dx & v &= \frac{1}{2} x^2. \end{align*}$

Then we have,

$\begin{align*} \int x \log x \, dx &= \int u \, dv \\ &= uv - \int v \, du \\ &= \frac{1}{2}x^2 \log |x| - \frac{1}{2} \int x \, dx \\ &= \frac{1}{2}x^2 \log |x| - \frac{1}{4} x^2 + C. \end{align*}$

2 comments

August 26, 2019 at 4:25 am

Anonymous says:

didnt get it

Reply
- November 30, 2023 at 10:04 am
  
  Anonymous says:
  
  Basically integration by parts. Assume x is the derivative of a function and compute the rest.
  
  Reply

Stumbling Robot

Find the integral of x log x

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