0
$\begingroup$

I would like to compute numerically the definite integral below using the Simpson rule:

$$\int\limits_0^1 \ln (x) \ln (1-x)\ dx$$

But I have difficulty trying to remove the singularity in this integral.

New contributor
Nweke Kizito is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
$\endgroup$
1

1 Answer 1

0
$\begingroup$

If you have a singularity with natural log, then your solution is imaginary. The indefinite integral of $\ln(x)\ln(x-1)$ is $$x((\ln(x-1)-1)\ln(x)-\ln(x-1)+2)+\ln(x-1)+ Li_2 (1-x)+C$$Using integral-calculator.com . Li_2 is $$\displaystyle\sum^{\infty}_{k=1} \frac{x^k}{k^2}$$ Now note that the natural log of -1 is $\pi i$, so our final answer is $$\pi i -2+\frac{\pi^2}{6}$$ Hope this helps :)

$\endgroup$

Your Answer

Nweke Kizito is a new contributor. Be nice, and check out our Code of Conduct.

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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