1.02y Partial fractions: decompose rational functions

8 Let \(\mathrm { f } ( x ) = \frac { 10 x + 9 } { ( 2 x + 1 ) ( 2 x + 3 ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 2 } \ln \frac { 9 } { 5 } + \frac { 1 } { 5 }\).

9 Let \(\mathrm { f } ( x ) = \frac { 2 x ( 5 - x ) } { ( 3 + x ) ( 1 - x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).

9 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 3 } + 6 x - 8 } { x \left( x ^ { 2 } + 2 \right) }\).

1. Express \(\mathrm { f } ( x )\) in the form \(A + \frac { B } { x } + \frac { C x + D } { x ^ { 2 } + 2 }\).
2. Show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 3 - \ln 4\).

9 Let \(\mathrm { f } ( x ) = \frac { x ( 6 - x ) } { ( 2 + x ) \left( 4 + x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

8 Let \(\mathrm { f } ( x ) = \frac { 12 + 12 x - 4 x ^ { 2 } } { ( 2 + x ) ( 3 - 2 x ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

6 Let \(f ( x ) = \frac { 6 + 7 x } { ( 2 - x ) \left( 1 + x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that, when \(x\) is sufficiently small for \(x ^ { 4 }\) and higher powers to be neglected, $$f ( x ) = 3 + 5 x - \frac { 1 } { 2 } x ^ { 2 } - \frac { 15 } { 4 } x ^ { 3 }$$

9 In an experiment to study the spread of a soil disease, an area of \(10 \mathrm {~m} ^ { 2 }\) of soil was exposed to infection. In a simple model, it is assumed that the infected area grows at a rate which is proportional to the product of the infected area and the uninfected area. Initially, \(5 \mathrm {~m} ^ { 2 }\) was infected and the rate of growth of the infected area was \(0.1 \mathrm {~m} ^ { 2 }\) per day. At time \(t\) days after the start of the experiment, an area \(a \mathrm {~m} ^ { 2 }\) is infected and an area \(( 10 - a ) \mathrm { m } ^ { 2 }\) is uninfected.

1. Show that \(\frac { \mathrm { d } a } { \mathrm {~d} t } = 0.004 a ( 10 - a )\).
2. By first expressing \(\frac { 1 } { a ( 10 - a ) }\) in partial fractions, solve this differential equation, obtaining an expression for \(t\) in terms of \(a\).
3. Find the time taken for \(90 \%\) of the soil area to become infected, according to this model.

8 An appropriate form for expressing \(\frac { 3 x } { ( x + 1 ) ( x - 2 ) }\) in partial fractions is $$\frac { A } { x + 1 } + \frac { B } { x - 2 }$$ where \(A\) and \(B\) are constants.

1. Without evaluating any constants, state appropriate forms for expressing the following in partial fractions:
   1. \(\frac { 4 x } { ( x + 4 ) \left( x ^ { 2 } + 3 \right) }\),
   2. \(\frac { 2 x + 1 } { ( x - 2 ) ( x + 2 ) ^ { 2 } }\).
2. Show that \(\int _ { 3 } ^ { 4 } \frac { 3 x } { ( x + 1 ) ( x - 2 ) } \mathrm { d } x = \ln 5\).

9

1. Express \(\frac { 3 x ^ { 2 } + x } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 3 x ^ { 2 } + x } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

8

1. The complex number \(z\) is given by \(z = \frac { 4 - 3 \mathrm { i } } { 1 - 2 \mathrm { i } }\).
   1. Express \(z\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
   2. Find the modulus and argument of \(z\).
2. Find the two square roots of the complex number 5-12i, giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

8 \includegraphics[max width=\textwidth, alt={}, center]{c687888e-bef0-4ea9-b5b3-e614028cc07c-3_654_805_274_671} An underground storage tank is being filled with liquid as shown in the diagram. Initially the tank is empty. At time \(t\) hours after filling begins, the volume of liquid is \(V \mathrm {~m} ^ { 3 }\) and the depth of liquid is \(h \mathrm {~m}\). It is given that \(V = \frac { 4 } { 3 } h ^ { 3 }\). The liquid is poured in at a rate of \(20 \mathrm {~m} ^ { 3 }\) per hour, but owing to leakage, liquid is lost at a rate proportional to \(h ^ { 2 }\). When \(h = 1 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = 4.95\).

1. Show that \(h\) satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 5 } { h ^ { 2 } } - \frac { 1 } { 20 } .$$
2. Verify that \(\frac { 20 h ^ { 2 } } { 100 - h ^ { 2 } } \equiv - 20 + \frac { 2000 } { ( 10 - h ) ( 10 + h ) }\).
3. Hence solve the differential equation in part (i), obtaining an expression for \(t\) in terms of \(h\).

8

1. Express \(\frac { 5 x + 3 } { ( x + 1 ) ^ { 2 } ( 3 x + 2 ) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 5 x + 3 } { ( x + 1 ) ^ { 2 } ( 3 x + 2 ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

8 Let \(f ( x ) = \frac { 12 + 8 x - x ^ { 2 } } { ( 2 - x ) \left( 4 + x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 2 - x } + \frac { B x + C } { 4 + x ^ { 2 } }\).
2. Show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \ln \left( \frac { 25 } { 2 } \right)\).

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - x ^ { 2 } + 4 x - a$$ where \(a\) is a constant. It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\).
2. When \(a\) has the value found in part (i), express \(\frac { 8 x - 13 } { \mathrm { p } ( x ) }\) in partial fractions.

9

1. Express \(\frac { 9 - 7 x + 8 x ^ { 2 } } { ( 3 - x ) \left( 1 + x ^ { 2 } \right) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 9 - 7 x + 8 x ^ { 2 } } { ( 3 - x ) \left( 1 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

8

1. Express \(\frac { 7 x ^ { 2 } + 8 } { ( 1 + x ) ^ { 2 } ( 2 - 3 x ) }\) in partial fractions.
2. Hence expand \(\frac { 7 x ^ { 2 } + 8 } { ( 1 + x ) ^ { 2 } ( 2 - 3 x ) }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

9 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } - 8 x + 9 } { ( 1 - x ) ( 2 - x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

10 By first using the substitution \(u = \mathrm { e } ^ { x }\), show that $$\int _ { 0 } ^ { \ln 4 } \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { 2 x } + 3 \mathrm { e } ^ { x } + 2 } \mathrm {~d} x = \ln \left( \frac { 8 } { 5 } \right)$$

7

1. Show that ( \(x + 1\) ) is a factor of \(4 x ^ { 3 } - x ^ { 2 } - 11 x - 6\).
2. Find \(\int \frac { 4 x ^ { 2 } + 9 x - 1 } { 4 x ^ { 3 } - x ^ { 2 } - 11 x - 6 } \mathrm {~d} x\).

8 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 2 } + x + 6 } { ( x + 2 ) \left( x ^ { 2 } + 4 \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

9 Let \(\mathrm { f } ( x ) = \frac { 6 x ^ { 2 } + 8 x + 9 } { ( 2 - x ) ( 3 + 2 x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence, showing all necessary working, show that \(\int _ { - 1 } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x = 1 + \frac { 1 } { 2 } \ln \left( \frac { 3 } { 4 } \right)\).

8 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + x + 6 } { x ^ { 2 } ( x + 2 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence, showing full working, show that the exact value of \(\int _ { 1 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x\) is \(\frac { 9 } { 4 }\).

8 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + x + 8 } { ( 2 x - 1 ) \left( x ^ { 2 } + 2 \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence, showing full working, find \(\int _ { 1 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x\), giving the answer in the form \(\ln c\), where \(c\) is an integer.

9 The variables \(x\) and \(t\) satisfy the differential equation \(5 \frac { \mathrm {~d} x } { \mathrm {~d} t } = ( 20 - x ) ( 40 - x )\). It is given that \(x = 10\) when \(t = 0\).

1. Using partial fractions, solve the differential equation, obtaining an expression for \(x\) in terms of \(t\). [9]
2. State what happens to the value of \(x\) when \(t\) becomes large.

4

1. By first expressing \(\frac { 1 } { r ^ { 2 } - 1 }\) in partial fractions, show that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ^ { 2 } - 1 } = \frac { 3 } { 4 } - \frac { a n + b } { 2 n ( n + 1 ) }$$ where \(a\) and \(b\) are integers to be found.
2. Deduce the value of \(\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ^ { 2 } - 1 }\).
3. Find the limit, as \(n \rightarrow \infty\), of \(\sum _ { r = n + 1 } ^ { 2 n } \frac { n } { r ^ { 2 } - 1 }\).