Not able to solve $\int\limits_1^n \frac{g(x)}{x^{p+1}} \mathrm dx $

Question

If $p=\frac{7}{8}$ then what should be the value of $\displaystyle\int\limits_1^n \frac{g(x)}{x^{p+1}} \mathrm dx $ when $$g(x) = x \log x \quad \text{or} \quad g(x) = \frac{x}{\log x}? $$

Wondering which way to proceed?

1. an algebraic substitution,
2. partial fractions,
3. integration by parts, or
4. reduction formulae.

Please don't suggest something like ("Learn basic Calculus first" etc).
Kindly help by solving if possible because I'm out of touch with calculus for nearly 15 yrs.

Answer 1

$$\frac{g(x)}{x^{p+1}}=\frac{x\log x}{x^{p+1}}=\frac{\log x}{x^p}$$

By parts:

$$u=\frac1{x^p}\;,\;\;u'=-\frac p{x^{p+1}}\\v'=\log x\;,\;\;v=x\log x-x$$

Thus:

$$\int\limits_1^n\frac{\log x}{x^p}dx=\left.\left(\frac{\log x}{x^{p-1}}-\frac1{x^{p-1}}\right)\right|_1^n+p\int\limits_1^n\frac{\log x}{x^p}dx-p\int\limits_1^n\frac1{x^p}dx\ldots\ldots$$