Question 3 - A-Level Maths

Exam Board	AQA
Module	C3 (Core Mathematics 3)
Year	2006
Session	January
Topic	Integration by Substitution

1. Given that $\mathrm { f } ( x ) = x ^ { 4 } + 2 x$, find $\mathrm { f } ^ { \prime } ( x )$.
2. Hence, or otherwise, find $\int \frac { 2 x ^ { 3 } + 1 } { x ^ { 4 } + 2 x } \mathrm {~d} x$.
1. Use the substitution $u = 2 x + 1$ to show that $$\int x \sqrt { 2 x + 1 } \mathrm {~d} x = \frac { 1 } { 4 } \int \left( u ^ { \frac { 3 } { 2 } } - u ^ { \frac { 1 } { 2 } } \right) \mathrm { d } u$$
2. Hence show that $\int _ { 0 } ^ { 4 } x \sqrt { 2 x + 1 } \mathrm {~d} x = 19.9$ correct to three significant figures.