Question 3 - A-Level Maths

Exam Board	Edexcel
Module	FP3 (Further Pure Mathematics 3)
Year	2011
Session	June
Topic	Integration using inverse trig and hyperbolic functions

Show that
1. $\int _ { 5 } ^ { 8 } \frac { 1 } { x ^ { 2 } - 10 x + 34 } \mathrm {~d} x = k \pi$, giving the value of the fraction $k$,
2. $\int _ { 5 } ^ { 8 } \frac { 1 } { \sqrt { } \left( x ^ { 2 } - 10 x + 34 \right) } \mathrm { d } x = \ln ( A + \sqrt { } n )$, giving the values of the integers $A$ and $n$.
$$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ^ { 2 } ( \ln x ) ^ { n } \mathrm {~d} x , \quad n \geqslant 0$$
Prove that, for $n \geqslant 1$, $$I _ { n } = \frac { \mathrm { e } ^ { 3 } } { 3 } - \frac { n } { 3 } I _ { n - 1 }$$
Find the exact value of $I _ { 3 }$.