Good morning everyone! ッ hru guys? R(n)∼2Pi_2product_(k=2; p_k|n)(p_k-1)/(p_k-2)int_2^n(dx)/(( lnx)^2),

Explanation is in the file

https://cutt.ly/kzjInUL

GOODMORNING im great you?

Explanation:   Answer In =

Z

(ln x)

n

dx

Integrate by parts with

u = (ln x)

n

dv

dx = 1

du

dx = n(ln x)

n−1 ×

1

x

v = x

Therefore

In = x(ln x)

n −

Z

dxn(ln x)

n−1

x

× x

= x(ln x)

n − n

Z

dx(ln x)

n−1 = x(ln x)

n − nIn−1

Hence In = x(ln x)

n − nIn−1 is the reduction formula.

If I4 =

Z e

1

(ln x)

4

dx then inserting the integration limits into the above expression we have, with n = 4.

I4 = [x(ln x)

4

]

e

1 − 4I3

I3 = [x(ln x)

3

]

e

1 − 3I − 2

I − 2 = [x(ln x)

2

]

e

1 − 2I1

I1 = [x(ln x)]e

1 − I0

What’s I0?

I − 0 =

Z e

1

(ln x)

0

dx =

Z e

1

dx = e − 1

Therefore

I1 = (e ln e − 1 ln 1) − (e − 1) (ln 1 = 0, ln e = 1)

= e − (e − 1) = 1

Hence I2 = [e(ln e)

2 − 1(ln 1)2

] − 2I1

= e − 2 × 1 = e − 2

Hence I3 = [e(ln e)

3 − 1(ln 1)3

] − 3I2

= e − 3 × (e − 2) = 6 − 2e

Hence I4 = [e(ln e)

4 − 1(ln 1)4

] − 4I3

= e − 4 × (6 − 2e) = 9e − 24

~hope this helps~

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