1.02y Partial fractions: decompose rational functions

6. $$f ( x ) = \frac { 15 - 17 x } { ( 2 + x ) ( 1 - 3 x ) ^ { 2 } } , \quad x \neq - 2 , \quad x \neq \frac { 1 } { 3 }$$

1. Find the values of the constants \(A , B\) and \(C\) such that $$\mathrm { f } ( x ) = \frac { A } { 2 + x } + \frac { B } { 1 - 3 x } + \frac { C } { ( 1 - 3 x ) ^ { 2 } }$$
2. Find the value of $$\int _ { - 1 } ^ { 0 } f ( x ) d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.
   6. continued

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

3. (a) Express \(\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } }\) as a sum of partial fractions.
(b) Hence find the series expansion of \(\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } } , | x | < \frac { 1 } { 4 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
3. continued

8. (a) Show that the substitution \(u = \sin x\) transforms the integral $$\int \frac { 6 } { \cos x ( 2 - \sin x ) } d x$$ into the integral $$\int \frac { 6 } { \left( 1 - u ^ { 2 } \right) ( 2 - u ) } \mathrm { d } u .$$ (b) Express \(\frac { 6 } { \left( 1 - u ^ { 2 } \right) ( 2 - u ) }\) in partial fractions.
(c) Hence, evaluate $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 6 } { \cos x ( 2 - \sin x ) } d x$$ giving your answer in the form \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are integers.
8. continued
8. continued

3. $$f ( x ) = \frac { 7 + 3 x + 2 x ^ { 2 } } { ( 1 - 2 x ) ( 1 + x ) ^ { 2 } } , \quad | x | > \frac { 1 } { 2 }$$

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that $$\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = p - \ln q$$ where \(p\) is rational and \(q\) is an integer.
   3. continued

2

1. Given that $$\frac { 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { A } { r ( r + 1 ) } + \frac { B } { ( r + 1 ) ( r + 2 ) }$$ show that \(A = \frac { 1 } { 2 }\) and find the value of \(B\).
2. Use the method of differences to find $$\sum _ { r = 10 } ^ { 98 } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$$ giving your answer as a rational number.

2

1. Express \(\frac { 1 } { r ( r + 2 ) }\) in partial fractions.
2. Use the method of differences to find $$\sum _ { r = 1 } ^ { 48 } \frac { 1 } { r ( r + 2 ) }$$ giving your answer as a rational number.

1

1. Express \(\frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\) in partial fractions.
2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\), expressing the result as a single fraction.

1. Show that

$$\int _ { 0 } ^ { \infty } \frac { 8 x - 12 } { \left( 2 x ^ { 2 } + 3 \right) ( x + 1 ) } \mathrm { d } x = \ln k$$ where \(k\) is a rational number to be found.

6

1. Find as a single algebraic fraction an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).
2. Determine the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).

7 In this question you must show detailed reasoning.
Find \(\int _ { 2 } ^ { 3 } \frac { x + 1 } { x ^ { 3 } - x ^ { 2 } + x - 1 } \mathrm {~d} x\), expressing your answer in the form \(a \ln b\) where \(a\) and \(b\) are rational numbers.

9

1. Express \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in three partial fractions.
2. Hence find the first three terms in the expansion of \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\).
3. State the set of values for which the expansion in part (ii) is valid.

11 In this question you must show detailed reasoning. A function f is given by \(\mathrm { f } ( x ) = \frac { x - 4 } { ( x + 2 ) ( x - 1 ) } + \frac { 3 x + 1 } { ( x + 3 ) ( x - 1 ) }\).

1. Show that \(\mathrm { f } ( x )\) can be written as \(\frac { 2 ( 2 x + 5 ) } { ( x + 2 ) ( x + 3 ) }\).
2. Given that \(\int _ { a } ^ { a + 4 } \mathrm { f } ( x ) \mathrm { d } x = 2 \ln 3\), find the value of the positive constant \(a\).

4 Express \(\frac { 5 x ^ { 2 } + x + 12 } { x ^ { 3 } + 4 x }\) in partial fractions.

3. The function \(f\) is defined by $$f : x \mapsto \frac { 5 x + 1 } { x ^ { 2 } + x - 2 } - \frac { 3 } { x + 2 } , x > 1$$

1. Show that \(\mathrm { f } ( x ) = \frac { 2 } { x - 1 } , x > 1\).
2. Find \(\mathrm { f } ^ { - 1 } ( x )\). The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } + 5 , \quad x \in \mathbb { R } .$$ (b) Solve \(\mathrm { fg } ( x ) = \frac { 1 } { 4 }\).

6. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\), where \(a\) and \(b\) are constants. The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.

1. Given that \(L\) and \(H\) meet, show that the \(x\)-coordinates of the points of intersection are the roots of the equation $$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$ Hence, given that \(L\) is a tangent to \(H\),
2. show that \(a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }\). The hyperbola \(H ^ { \prime }\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1\).
3. Find the equations of the tangents to \(H ^ { \prime }\) which pass through the point \(( 1,4 )\).

5

1. 1. Obtain the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
   2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of \(x\).
2. Obtain the binomial expansion of \(\frac { 1 } { ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
3. Given that \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) can be written in the form \(\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }\), find the values of \(A , B\) and \(C\).
4. Hence find the binomial expansion of \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).

5

1. 1. Obtain the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
      (2 marks)
   2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of \(x\).
2. Obtain the binomial expansion of \(\frac { 1 } { ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
3. Given that \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) can be written in the form \(\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }\), find the values of \(A , B\) and \(C\).
4. Hence find the binomial expansion of \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).

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 \(\int \frac { 6 x - 5 } { 4 x ^ { 2 } - 25 } \mathrm {~d} x\).

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. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 18 x + 8\).
   1. Use the Factor Theorem to show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
   2. Write \(\mathrm { f } ( x )\) in the form \(( 2 x - 1 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
   3. Simplify the algebraic fraction \(\frac { 4 x ^ { 2 } + 16 x } { 2 x ^ { 3 } + 3 x ^ { 2 } - 18 x + 8 }\).
2. Express the algebraic fraction \(\frac { 2 x ^ { 2 } } { ( x + 5 ) ( x - 3 ) }\) in the form \(A + \frac { B + C x } { ( x + 5 ) ( x - 3 ) }\), where \(A , B\) and \(C\) are integers.

3

1. 1. Express \(\frac { 2 x + 7 } { x + 2 }\) in the form \(A + \frac { B } { x + 2 }\), where \(A\) and \(B\) are integers. (2 marks)
   2. Hence find \(\int \frac { 2 x + 7 } { x + 2 } \mathrm {~d} x\).
2. 1. Express \(\frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } }\) in the form \(\frac { P } { 1 + 3 x } + \frac { Q } { 5 - x } + \frac { R } { ( 5 - x ) ^ { 2 } }\), where \(P , Q\) and \(R\) are constants.
   2. Hence find \(\int \frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } } \mathrm {~d} x\).