Problem with improper integral with parameter

Question

I am supposed to calculate following integral: $$\int_0^1 \frac{1}{x\ln ^{p}x}dx$$

What I did was, that I substitute $\ln x=t$ and then I got: $$\int_{-\infty}^{0}\frac{1}{t^{p}}dt$$

But I do not know if it is right. What should I do next?

Answer 1

Correct, then it's simply an integral of a polynomial:

$$\int t^{-p}dt=\frac{t^{-p+1}}{-p+1}$$

P.S.: As @Naji notices, when $p=1$, then integral looks different:

$$\int t^{-1}dt=\ln{|t|}$$