How do I find the integral $int(x*ln(x))dx$ ?

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How do I find the integral $int(x*ln(x))dx$ ?

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Answer 1 · 2014-08-18T03:03:54

We will use integration by parts.

Remember the IBP's formula, which is

$int u dv = uv - int v du$

Let $u = ln x$ , and $dv = x dx$ . We have chosen these values because we know that the derivative of $ln x$ is equal to $1/x$ , meaning that instead of integrating something complex (a natural logarithm) we now will end up integrating something pretty easy. (a polynomial)

Thus, $du = 1/x dx$ , and $v = x^2 / 2$ .

Plugging into the IBP's formula gives us:

$int x ln x dx = (x^2 ln x)/2 - int x^2 / (2x) dx$

An $x$ will cancel off from the new integrand:

$int x ln x dx = (x^2 ln x)/2 - int x / 2 dx$

The solution is now easily found using the power rule. Don't forget the constant of integration:

$int x ln x dx = (x^2 ln x)/2 - x^2 / 4 + C$

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How do I find the integral $int(x*ln(x))dx$ ?

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