# Integral of Natural Logs

I had this problem on an integral test today. I tried using u substitution but to no avail.

Integral: $\int (1+\ln(x))x\cdot \ln(x)dx$.

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Just to check, do you know how to do $\int \ln(x) \, dx$? – Calvin Lin Feb 6 '13 at 23:58
There is a $u$ substitution that makes everything doable in one step. I don't want to give it away since finding the substitution is the whole deal, but keep trying. Think about what the ideal $u$ substitution would look like. – Christopher A. Wong Feb 7 '13 at 0:03
I edited your post to correctly display the math symbols. You might want to have a look to learn how to do it yourself. – Ittay Weiss Feb 7 '13 at 0:23

## 2 Answers

Two ways to do it. First of all, you can make the substitution $x=e^u$, which reduces the integral to

$$\int(1+u)ue^{2u}du$$

Two iterations of integration by parts will eliminate the polynomial, giving something you can integrate directly.

As to the second method, I assume when learning integration by parts you might have seen

$$\int\ln xdx=x\ln x+x$$

Based on this, the substitution $u=x\ln x+x$ reduces the integral to $\int udu$.

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Multiply out. We want the integrals of $x\ln x$ and $x\ln^2 x$.

For the first integral, use integration by parts, $u=\ln x$, $dv=x\,dx$. When you do the details, you will end up needing to integrate $\frac{x}{2}$.

For the second integral, again use integration by parts, letting $u=\ln^2 x$, $dv=x\,dx$. You will not be quite finished, but there will be clear progress.

Another way: Let $u=x\ln x$. Then $du=(x(1/x)+\ln x)\,dx=(1+\ln x)\,dx$.

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