Algebraic manipulation before substitution

2 Use the substitution \(u = 3 x + 1\) to find \(\int \frac { 3 x } { 3 x + 1 } \mathrm {~d} x\).

5 Find the exact value of \(\int _ { 0 } ^ { 6 } \frac { x ( x + 1 ) } { x ^ { 2 } + 4 } \mathrm {~d} x\).

3. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Given that \(k\) is a positive constant,

1. find $$\int \frac { 9 x } { 3 x ^ { 2 } + k } d x$$ Given also that $$\int _ { 2 } ^ { 5 } \frac { 9 x } { 3 x ^ { 2 } + k } \mathrm {~d} x = \ln 8$$
2. find the value of \(k\)

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$f ( x ) = \frac { 2 x ^ { 3 } - 4 x - 15 } { x ^ { 2 } + 3 x + 4 }$$

1. Show that $$f ( x ) \equiv A x + B + \frac { C ( 2 x + 3 ) } { x ^ { 2 } + 3 x + 4 }$$ where \(A , B\) and \(C\) are integers to be found.
2. Hence, find $$\int _ { 3 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.

5. (i) Find $$\int \left( ( 3 x + 5 ) ^ { 9 } + \mathrm { e } ^ { 5 x } \right) \mathrm { d } x$$ (ii) Given that \(b\) is a constant greater than 2 , and $$\int _ { 2 } ^ { b } \frac { x } { x ^ { 2 } + 5 } \mathrm {~d} x = \ln ( \sqrt { 6 } )$$ use integration to find the value of \(b\).

5. $$4 x ^ { 2 } + 4 x + 17 \equiv ( 2 x + p ) ^ { 2 } + q$$ where \(p\) and \(q\) are integers.

1. Determine the value of \(p\) and the value of \(q\) Given that $$\frac { 8 x + 5 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } \equiv \frac { 1 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } + \frac { A x + B } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } }$$ where \(A\) and \(B\) are integers,
2. write down the value of \(A\) and the value of \(B\)
3. Hence use algebraic integration to show that $$\int _ { \frac { 1 } { 3 } } ^ { 1 } \frac { 8 x + 5 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } \mathrm {~d} x = k + \frac { 1 } { 2 } \ln k$$ where \(k\) is a rational number to be determined.

5 In this question, \(I\) denotes the definite integral \(\int _ { 2 } ^ { 5 } \frac { 5 - x } { 2 + \sqrt { x - 1 } } \mathrm {~d} x\). The value of \(I\) is to be found using two different methods.

1. Show that the substitution \(u = \sqrt { x - 1 }\) transforms \(I\) to \(\int _ { 1 } ^ { 2 } \left( 4 u - 2 u ^ { 2 } \right) \mathrm { d } u\) and hence find the exact value of \(I\).
2. (a) Simplify \(( 2 + \sqrt { x - 1 } ) ( 2 - \sqrt { x - 1 } )\).
   (b) By first multiplying the numerator and denominator of \(\frac { 5 - x } { 2 + \sqrt { x - 1 } }\) by \(2 - \sqrt { x - 1 }\), find the exact value of \(I\).

6 By using the substitution \(u = x - 2\), or otherwise, find the exact value of $$\int _ { - 1 } ^ { 5 } \frac { \mathrm {~d} x } { \sqrt { 32 + 4 x - x ^ { 2 } } }$$

6

1. Given that \(y = ( x - 2 ) \sqrt { 5 + 4 x - x ^ { 2 } } + 9 \sin ^ { - 1 } \left( \frac { x - 2 } { 3 } \right)\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \sqrt { 5 + 4 x - x ^ { 2 } }$$ where \(k\) is an integer.
2. Hence show that $$\int _ { 2 } ^ { \frac { 7 } { 2 } } \sqrt { 5 + 4 x - x ^ { 2 } } \mathrm {~d} x = p \sqrt { 3 } + q \pi$$ where \(p\) and \(q\) are rational numbers.
   [0pt] [3 marks]