Find antiderivative of $ln(x)^y$ for any real y

Question

What is this antiderivative? I have tested several values of $y$ in an online antiderivative calculator, but it's not clear how they are related. Here $y$ is fixed and I want the antiderivative with respect to $x$. I am particularly interested in the value of the integral with respect to $x$ over $(0, 1)$, for values of $y$ for which $ln(x)^y$ is defined there.

Answer 1

We set:

$$\text{I}_y=\int\ln^y(x)\space\text{d}x$$

Now, by integration by parts:

$$\text{I}_y=x\log^y(x)-y\int\log^{y-1}(x)\space\text{d}x=x\log^y(x)-y\cdot\text{I}_{y-1}$$

So, by induction we can find that:

• When $y=0$: $$\text{I}_0=x+\text{C}$$
• When $y\ge1$: $$\text{I}_y=x\ln^y(x)-y\cdot\text{I}_{y-1}$$

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Mathematica gives:

$$\text{I}_y=\frac{\ln^y(x)\Gamma\left(1+y,-\ln(x)\right)}{\left(-\ln(x)\right)^y}+\text{C}$$

Where $\Gamma(a,x)$ is the incomplete gamma function.