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Using Riemann Zeta Function in order to find the value of an integral

I am trying to solve this question and I need some help. I've never used the Riemann Zeta Function so I need some help in order to find the answer:

Find the value of

$$\int_2^5 \frac\log^3(x)x^2\log(x)dx$$

A:

Let $I(a) = \int_2^5 \frac\log^3(x)x^2\log(x) dx$. Note that if you integrate by parts then 

$$I(a) = -3 \log^2(5) + 3 I(2) - 2 I(1).$$

We may use the fact that 

$$I(a) = \int_2^a \frac\log^3(x)x^2 dx + \int_a^5 \frac\log^3(x)x^2 dx = \int_2^a \frac\log^3(x)x dx + \int_a^5 \frac\log^3(x)x dx - 2 I(a),$$

to see that

$$I(5) = \int_2^5 \frac\log^3(x)x dx - 2 I(5).$$

Here we recognize that $I(1) = \zeta(3)$, and $I(2) = \zeta(4)$. In the end, 

$$I(5) = 2 \zeta(3) + 2 \zeta(4) - 2 \zeta(5) = \frac25 \zeta(3) - 2\zeta(5).$$

Alternatively, this is a partial fraction decomposition with denominator $x^2 \log(x)$, which can be written 

$$\frac1x^2 \log(x) = \frac13x^2 + \frac2x^2 \log(x) - \frac2x^2$$

then,