1.02y Partial fractions: decompose rational functions

1. The function f is defined by

$$\mathrm { f } ( x ) = \frac { 6 } { 2 x + 5 } + \frac { 2 } { 2 x - 5 } + \frac { 60 } { 4 x ^ { 2 } - 25 } , \quad x > 4$$

1. Show that \(\mathrm { f } ( x ) = \frac { A } { B x + C }\) where \(A , B\) and \(C\) are constants to be found.
2. Find \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.

4. $$\mathrm { f } ( x ) = x + \frac { 3 } { x - 1 } - \frac { 12 } { x ^ { 2 } + 2 x - 3 } , x \in \mathbb { R } , x > 1$$

1. Show that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 x + 3 } { x + 3 }\).
2. Solve the equation \(\mathrm { f } ^ { \prime } ( x ) = \frac { 22 } { 25 }\).

10. (a) Write \(\frac { 1 } { ( H - 5 ) ( H + 3 ) }\) in partial fraction form. The depth of water in a storage tank is being monitored.
The depth of water in the tank, \(H\) metres, is modelled by the differential equation $$\frac { \mathrm { d } H } { \mathrm {~d} t } = - \frac { ( H - 5 ) ( H + 3 ) } { 40 }$$ where \(t\) is the time, in days, from when monitoring began.
Given that the initial depth of water in the tank was 13 m ,
(b) solve the differential equation to show that $$H = \frac { 10 + 3 \mathrm { e } ^ { - 0.2 t } } { 2 - \mathrm { e } ^ { - 0.2 t } }$$ (c) Hence find the time taken for the depth of water in the tank to fall to 8 m .
(Solutions relying entirely on calculator technology are not acceptable.) According to the model, the depth of water in the tank will eventually fall to \(k\) metres.
(d) State the value of the constant \(k\).

4. $$\mathrm { f } ( x ) = \frac { 4 - 4 x } { x ( x - 2 ) ^ { 2 } } \quad x > 2$$

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\)
3. Find $$\int _ { 3 } ^ { 5 } f ( x ) d x$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are rational numbers to be found.

3. The curve \(C\) has parametric equations $$x = 3 + 2 \sin t \quad y = \frac { 6 } { 7 + \cos 2 t } \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$

1. Show that \(C\) has Cartesian equation $$y = \frac { 12 } { ( 7 - x ) ( 1 + x ) } \quad p \leqslant x \leqslant q$$ where \(p\) and \(q\) are constants to be found.
2. Hence, find a Cartesian equation for \(C\) in the form $$y = \frac { a } { x + b } + \frac { c } { x + d } \quad p \leqslant x \leqslant q$$ where \(a , b , c\) and \(d\) are constants.

1. $$f ( x ) = \frac { 5 x + 10 } { ( 1 - x ) ( 2 + 3 x ) }$$

1. Write \(\mathrm { f } ( x )\) in partial fraction form.
2. 1. Hence find, in ascending powers of \(x\) up to and including the terms in \(x ^ { 2 }\), the binomial series expansion of \(\mathrm { f } ( x )\). Give each coefficient as a simplified fraction.
   2. Find the range of values of \(x\) for which this expansion is valid.

1. Given that

$$\frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \equiv \frac { A } { x - 2 } + \frac { B } { 2 x + 1 } + \frac { C } { ( 2 x + 1 ) ^ { 2 } }$$

1. find the values of the constants \(A , B\) and \(C\).
2. Hence find the exact value of $$\int _ { 7 } ^ { 12 } \frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \mathrm {~d} x$$ giving your answer in the form \(p \ln q + r\) where \(p\), \(q\) and \(r\) are rational numbers.

5. $$f ( x ) = \frac { 3 x ^ { 2 } + 16 } { ( 1 - 3 x ) ( 2 + x ) ^ { 2 } } = \frac { A } { ( 1 - 3 x ) } + \frac { B } { ( 2 + x ) } + \frac { C } { ( 2 + x ) ^ { 2 } } , \quad | x | < \frac { 1 } { 3 } .$$

1. Find the values of \(A\) and \(C\) and show that \(B = 0\).
2. Hence, or otherwise, find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Simplify each term.

4. (a) Express \(\frac { 2 x - 1 } { ( x - 1 ) ( 2 x - 3 ) }\) in partial fractions.
(b) Given that \(x \geqslant 2\), find the general solution of the differential equation $$( 2 x - 3 ) ( x - 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x - 1 ) y$$ (c) Hence find the particular solution of this differential equation that satisfies \(y = 10\) at \(x = 2\), giving your answer in the form \(y = \mathrm { f } ( x )\).

3. $$f ( x ) = \frac { 27 x ^ { 2 } + 32 x + 16 } { ( 3 x + 2 ) ^ { 2 } ( 1 - x ) } , \quad | x | < \frac { 2 } { 3 }$$ Given that \(\mathrm { f } ( x )\) can be expressed in the form $$f ( x ) = \frac { A } { ( 3 x + 2 ) } + \frac { B } { ( 3 x + 2 ) ^ { 2 } } + \frac { C } { ( 1 - x ) }$$

1. find the values of \(B\) and \(C\) and show that \(A = 0\).
2. Hence, or otherwise, find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). Simplify each term.
3. Find the percentage error made in using the series expansion in part (b) to estimate the value of \(\mathrm { f } ( 0.2 )\). Give your answer to 2 significant figures. \section*{LU}

3. (a) Express \(\frac { 5 } { ( x - 1 ) ( 3 x + 2 ) }\) in partial fractions.
(b) Hence find \(\int \frac { 5 } { ( x - 1 ) ( 3 x + 2 ) } \mathrm { d } x\), where \(x > 1\).
(c) Find the particular solution of the differential equation $$( x - 1 ) ( 3 x + 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = 5 y , \quad x > 1$$ for which \(y = 8\) at \(x = 2\). Give your answer in the form \(y = \mathrm { f } ( x )\).

1. (a) Express \(\frac { 1 } { P ( 5 - P ) }\) in partial fractions.

A team of conservationists is studying the population of meerkats on a nature reserve. The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 15 } P ( 5 - P ) , \quad t \geqslant 0$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that when \(t = 0 , P = 1\),
(b) solve the differential equation, giving your answer in the form, $$P = \frac { a } { b + c \mathrm { e } ^ { - \frac { 1 } { 3 } t } }$$ where \(a\), \(b\) and \(c\) are integers.
(c) Hence show that the population cannot exceed 5000

3. Express \(\frac { 9 x ^ { 2 } + 20 x - 10 } { ( x + 2 ) ( 3 x - 1 ) }\) in partial fractions.

2. (i) Find $$\int x \cos \left( \frac { x } { 2 } \right) \mathrm { d } x$$ (ii) (a) Express \(\frac { 1 } { x ^ { 2 } ( 1 - 3 x ) }\) in partial fractions.
(b) Hence find, for \(0 < x < \frac { 1 } { 3 }\) $$\int \frac { 1 } { x ^ { 2 } ( 1 - 3 x ) } \mathrm { d } x$$

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.

2. $$f ( x ) = \frac { 3 x - 1 } { ( 1 - 2 x ) ^ { 2 } } , \quad | x | < \frac { 1 } { 2 }$$ Given that, for \(x \neq \frac { 1 } { 2 } , \quad \frac { 3 x - 1 } { ( 1 - 2 x ) ^ { 2 } } = \frac { A } { ( 1 - 2 x ) } + \frac { B } { ( 1 - 2 x ) ^ { 2 } } , \quad\) where \(A\) and \(B\) are constants,

1. find the values of \(A\) and \(B\).
2. Hence, or otherwise, find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying each term.
   (6)

4. $$\frac { 2 \left( 4 x ^ { 2 } + 1 \right) } { ( 2 x + 1 ) ( 2 x - 1 ) } \equiv A + \frac { B } { ( 2 x + 1 ) } + \frac { C } { ( 2 x - 1 ) } .$$

1. Find the values of the constants \(A , B\) and \(C\).
2. Hence show that the exact value of \(\int _ { 1 } ^ { 2 } \frac { 2 \left( 4 x ^ { 2 } + 1 \right) } { ( 2 x + 1 ) ( 2 x - 1 ) } \mathrm { d } x\) is \(2 + \ln k\), giving the value of the constant \(k\).

7. (a) Express \(\frac { 2 } { 4 - y ^ { 2 } }\) in partial fractions.
(b) Hence obtain the solution of $$2 \cot x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 4 - y ^ { 2 } \right)$$ for which \(y = 0\) at \(x = \frac { \pi } { 3 }\), giving your answer in the form \(\sec ^ { 2 } x = \mathrm { g } ( y )\).

3. $$\mathrm { f } ( x ) = \frac { 4 - 2 x } { ( 2 x + 1 ) ( x + 1 ) ( x + 3 ) } = \frac { A } { 2 x + 1 } + \frac { B } { x + 1 } + \frac { C } { x + 3 }$$

1. Find the values of the constants \(A , B\) and \(C\).
2. 1. Hence find \(\int f ( x ) \mathrm { d } x\).
   2. Find \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\) in the form \(\ln k\), where \(k\) is a constant.

5. $$\frac { 2 x ^ { 2 } + 5 x - 10 } { ( x - 1 ) ( x + 2 ) } \equiv A + \frac { B } { x - 1 } + \frac { C } { x + 2 }$$

1. Find the values of the constants \(A , B\) and \(C\).
2. Hence, or otherwise, expand \(\frac { 2 x ^ { 2 } + 5 x - 10 } { ( x - 1 ) ( x + 2 ) }\) in ascending powers of \(x\), as far as the term in \(x ^ { 2 }\). Give each coefficient as a simplified fraction.

1. $$\frac { 9 x ^ { 2 } } { ( x - 1 ) ^ { 2 } ( 2 x + 1 ) } = \frac { A } { ( x - 1 ) } + \frac { B } { ( x - 1 ) ^ { 2 } } + \frac { C } { ( 2 x + 1 ) }$$ Find the values of the constants \(A , B\) and \(C\).

1. $$\mathrm { f } ( x ) = \frac { 1 } { x ( 3 x - 1 ) ^ { 2 } } = \frac { A } { x } + \frac { B } { ( 3 x - 1 ) } + \frac { C } { ( 3 x - 1 ) ^ { 2 } }$$

1. Find the values of the constants \(A , B\) and \(C\).
2. 1. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
   2. Find \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\), leaving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants. 1 \(f ( x ) = \frac { 1 } { x ( 3 x - 1 ) ^ { 2 } } = \frac { A } { x } + \frac { } { ( 3 x }\)
      1. Find the values of the constants \(A , B\) and \(C\).

1. Express in partial fractions

$$\frac { 5 x + 3 } { ( 2 x + 1 ) ( x + 1 ) ^ { 2 } }$$

1. (a) Use the substitution \(x = u ^ { 2 } , u > 0\), to show that

$$\int \frac { 1 } { x ( 2 \sqrt { x } - 1 ) } \mathrm { d } x = \int \frac { 2 } { u ( 2 u - 1 ) } \mathrm { d } u$$ (b) Hence show that $$\int _ { 1 } ^ { 9 } \frac { 1 } { x ( 2 \sqrt { x } - 1 ) } \mathrm { d } x = 2 \ln \left( \frac { a } { b } \right)$$ where \(a\) and \(b\) are integers to be determined.

4. (a) Express \(\frac { 25 } { x ^ { 2 } ( 2 x + 1 ) }\) in partial fractions. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = \frac { 5 } { x \sqrt { } ( 2 x + 1 ) } , x > 0\) The finite region \(R\) is bounded by the curve \(C\), the \(x\)-axis, the line with equation \(x = 1\) and the line with equation \(x = 4\) This region is shown shaded in Figure 2 The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
(b) Use calculus to find the exact volume of the solid of revolution generated, giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are constants.