Partial fractions with algebraic division first

10

1. Use algebraic division to express \(\frac { x ^ { 3 } - 2 x ^ { 2 } - 4 x + 13 } { x ^ { 2 } - x - 6 }\) in the form \(A x + B + \frac { C x + D } { x ^ { 2 } - x - 6 }\), where \(A , B , C\) and \(D\) are constants.
2. Hence find \(\int _ { 4 } ^ { 6 } \frac { x ^ { 3 } - 2 x ^ { 2 } - 4 x + 13 } { x ^ { 2 } - x - 6 } \mathrm {~d} x\), giving your answer in the form \(a + \ln b\).

1

1. Express \(\frac { 2 x + 3 } { 4 x ^ { 2 } - 1 }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { 2 x + 1 }\), where \(A\) and \(B\) are integers. (3 marks)
2. Express \(\frac { 12 x ^ { 3 } - 7 x - 6 } { 4 x ^ { 2 } - 1 }\) in the form \(C x + \frac { D ( 2 x + 3 ) } { 4 x ^ { 2 } - 1 }\), where \(C\) and \(D\) are integers.
   (3 marks)
3. Evaluate \(\int _ { 1 } ^ { 2 } \frac { 12 x ^ { 3 } - 7 x - 6 } { 4 x ^ { 2 } - 1 } \mathrm {~d} x\), giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational numbers.
   (5 marks)

3

1. 1. Express \(\frac { 7 x - 3 } { ( x + 1 ) ( 3 x - 2 ) }\) in the form \(\frac { A } { x + 1 } + \frac { B } { 3 x - 2 }\).
   2. Hence find \(\int \frac { 7 x - 3 } { ( x + 1 ) ( 3 x - 2 ) } \mathrm { d } x\).
2. Express \(\frac { 6 x ^ { 2 } + x + 2 } { 2 x ^ { 2 } - x + 1 }\) in the form \(P + \frac { Q x + R } { 2 x ^ { 2 } - x + 1 }\).

1

1. 1. Express \(\frac { 5 x - 6 } { x ( x - 3 ) }\) in the form \(\frac { A } { x } + \frac { B } { x - 3 }\).
      (2 marks)
   2. Find \(\int \frac { 5 x - 6 } { x ( x - 3 ) } \mathrm { d } x\).
      (2 marks)
2. 1. Given that $$4 x ^ { 3 } + 5 x - 2 = ( 2 x + 1 ) \left( 2 x ^ { 2 } + p x + q \right) + r$$ find the values of the constants \(p , q\) and \(r\).
   2. Find \(\int \frac { 4 x ^ { 3 } + 5 x - 2 } { 2 x + 1 } \mathrm {~d} x\).

1

1. 1. Express \(\frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) }\) in the form \(\frac { A } { 2 + x } + \frac { B } { 1 - 3 x }\), where \(A\) and \(B\) are integers.
      (3 marks)
   2. Hence show that \(\int _ { - 1 } ^ { 0 } \frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) } \mathrm { d } x = p \ln 2\), where \(p\) is rational.
      (4 marks)
2. 1. Given that \(\frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }\) can be written as \(C + \frac { 5 - 8 x } { 2 - 5 x - 3 x ^ { 2 } }\), find the value of \(C\).
      (1 mark)
   2. Hence find the exact value of the area of the region bounded by the curve \(y = \frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }\), the \(x\)-axis and the lines \(x = - 1\) and \(x = 0\). You may assume that \(y > 0\) when \(- 1 \leqslant x \leqslant 0\).

3. (a) Find the values of the constants \(A , B , C\) and \(D\) such that $$\frac { 2 x ^ { 3 } - 5 x ^ { 2 } + 6 } { x ^ { 2 } - 3 x } \equiv A x + B + \frac { C } { x } + \frac { D } { x - 3 } .$$ (b) Evaluate $$\int _ { 1 } ^ { 2 } \frac { 2 x ^ { 3 } - 5 x ^ { 2 } + 6 } { x ^ { 2 } - 3 x } \mathrm {~d} x$$ giving your answer in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers.
3. continued

4

1. 1. Express \(\frac { 3 x - 5 } { x - 3 }\) in the form \(A + \frac { B } { x - 3 }\), where \(A\) and \(B\) are integers. (2 marks)
   2. Hence find \(\int \frac { 3 x - 5 } { x - 3 } \mathrm {~d} x\).
      (2 marks)
2. 1. Express \(\frac { 6 x - 5 } { 4 x ^ { 2 } - 25 }\) in the form \(\frac { P } { 2 x + 5 } + \frac { Q } { 2 x - 5 }\), where \(P\) and \(Q\) are integers.
      (3 marks)
   2. Hence find \(\int \frac { 6 x - 5 } { 4 x ^ { 2 } - 25 } \mathrm {~d} x\).

4 The expression \(\frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) }\) can be written in the form \(2 + \frac { A } { x + 1 } + \frac { B } { 5 x - 1 }\), where \(A\) and \(B\) are constants.

1. Find the values of \(A\) and \(B\).
2. Hence find \(\int \frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) } \mathrm { d } x\).

3

1. Given that \(\frac { 9 x ^ { 2 } - 6 x + 5 } { ( 3 x - 1 ) ( x - 1 ) }\) can be written in the form \(3 + \frac { A } { 3 x - 1 } + \frac { B } { x - 1 }\), where \(A\) and \(B\) are integers, find the values of \(A\) and \(B\).
2. Hence, or otherwise, find \(\int \frac { 9 x ^ { 2 } - 6 x + 5 } { ( 3 x - 1 ) ( x - 1 ) } \mathrm { d } x\).