Partial fractions with linear factors only

2

1. Express \(\frac { x } { ( x + 1 ) ( x + 2 ) }\) in partial fractions.
2. Hence find \(\int \frac { x } { ( x + 1 ) ( x + 2 ) } \mathrm { d } x\).

3

1. Express \(\frac { 3 - 2 x } { x ( 3 - x ) }\) in partial fractions.
2. Show that \(\int _ { 1 } ^ { 2 } \frac { 3 - 2 x } { x ( 3 - x ) } \mathrm { d } x = 0\).
3. What does the result of part (ii) indicate about the graph of \(y = \frac { 3 - 2 x } { x ( 3 - x ) }\) between \(x = 1\) and \(x = 2\) ?

5

1. Express \(\frac { 1 + x } { ( 1 - x ) ( 1 - 2 x ) }\) in partial fractions.
2. Hence find \(\int _ { 2 } ^ { 3 } \frac { 1 + x } { ( 1 - x ) ( 1 - 2 x ) } \mathrm { d } x\).

9. $$f ( x ) = \frac { 8 - x } { ( 1 + x ) ( 2 - x ) } , \quad | x | < 1$$

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x = \ln k$$ where \(k\) is an integer to be found.
3. Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.

4. (i) Express \(\frac { 3 x + 6 } { 3 x - x ^ { 2 } }\) in partial fractions.
(ii) Evaluate \(\int _ { 1 } ^ { 2 } \frac { 3 x + 6 } { 3 x - x ^ { 2 } } \mathrm {~d} x\).

6. $$f ( x ) = \frac { 1 + 3 x } { ( 1 - x ) ( 1 - 3 x ) } , \quad | x | < \frac { 1 } { 3 }$$

1. Find the values of the constants \(A\) and \(B\) such that $$\mathrm { f } ( x ) = \frac { A } { 1 - x } + \frac { B } { 1 - 3 x }$$
2. Evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } } f ( x ) d x$$ giving your answer as a single logarithm.
3. Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.

3. (i) Express \(\frac { x + 11 } { ( x + 4 ) ( x - 3 ) }\) as a sum of partial fractions.
(ii) Evaluate $$\int _ { 0 } ^ { 2 } \frac { x + 11 } { ( x + 4 ) ( x - 3 ) } d x$$ giving your answer in the form \(\ln k\), where \(k\) is an exact simplified fraction.

1 Using partial fractions, find \(\int \frac { x } { ( x + 1 ) ( 2 x + 1 ) } \mathrm { d } x\).

1. Express \(\cos \theta + \sqrt { 3 } \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(\alpha\) is acute, expressing \(\alpha\) in terms of \(\pi\).
2. Write down the derivative of \(\tan \theta\). Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { ( \cos \theta + \sqrt { 3 } \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = \frac { \sqrt { 3 } } { 4 }\).

2 Using partial fractions, find \(\int \frac { x } { ( x + 1 ) ( 2 x + 1 ) } \mathrm { d } x\).
[0pt] [7]

4

1. Express \(\frac { 1 } { ( x - 1 ) ( x + 2 ) }\) in partial fractions
   [0pt] [2]
2. In this question you must show detailed reasoning. Hence find \(\int _ { 2 } ^ { 3 } \frac { 1 } { ( x - 1 ) ( x + 2 ) } \mathrm { d } x\).
   Give your answer in its simplest form.

1 It is given that \(\mathrm { f } ( x ) = \frac { 19 x - 2 } { ( 5 - x ) ( 1 + 6 x ) }\) can be expressed as \(\frac { A } { 5 - x } + \frac { B } { 1 + 6 x }\), where \(A\) and \(B\) are integers.

1. Find the values of \(A\) and \(B\).
2. Hence show that \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = k \ln 5\), where \(k\) is a rational number.
   [0pt] [6 marks]

7. $$f ( x ) = \frac { 8 - x } { ( 1 + x ) ( 2 - x ) } , \quad | x | < 1$$

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x = \ln k$$ where \(k\) is an integer to be found.
3. Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
   7. continued
   7. continued

1. (a) Express \(\frac { x + 11 } { ( x + 4 ) ( x - 3 ) }\) as a sum of partial fractions.
   (b) Evaluate

$$\int _ { 0 } ^ { 2 } \frac { x + 11 } { ( x + 4 ) ( x - 3 ) } d x$$ giving your answer in the form \(\ln k\), where \(k\) is an exact simplified fraction. (5)
3. continued

6.

1. Express \(\frac { x } { ( x + 1 ) ( x + 2 ) }\) in partial fractions.
2. Hence find \(\int \frac { x } { ( x + 1 ) ( x + 2 ) } \mathrm { d } x\).

3. (a) Express \(\frac { 5 x + 3 } { ( 2 x - 3 ) ( x + 2 ) }\) in partial fractions.
(b) Hence find the exact value of \(\int _ { 2 } ^ { 6 } \frac { 5 x + 3 } { ( 2 x - 3 ) ( x + 2 ) } \mathrm { d } x\), giving your answer as a single logarithm.

1

1. Given that \(\frac { 3 } { 9 - x ^ { 2 } }\) can be expressed in the form \(k \left( \frac { 1 } { 3 + x } + \frac { 1 } { 3 - x } \right)\), find the value of the rational number \(k\).
2. Show that \(\int _ { 1 } ^ { 2 } \frac { 3 } { 9 - x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.

2

1. Express \(\frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) }\) in the form \(\frac { A } { x + 3 } + \frac { B } { 2 x - 1 }\).
2. Hence find \(\int \frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) } \mathrm { d } x\).

3

1. Express \(\frac { 1 } { ( x + 2 ) ( x + 3 ) }\) in partial fractions.
2. Find \(\int \frac { 1 } { ( x + 2 ) ( x + 3 ) } \mathrm { d } x\) in the form \(\ln ( \mathrm { f } ( x ) ) + c\), where \(c\) is the constant of integration and \(\mathrm { f } ( x )\) is a function to be determined.