Partial Fractions

1 Express \(\frac { 2 } { x - 1 } + \frac { 5 } { 2 x + 1 }\) as a single fraction.

5 A curve has equation $$y = \frac { x } { x ^ { 2 } - 1 }$$

1. Write down the equations of the three asymptotes to the curve.
2. Sketch the curve.
   (You are given that the curve has no stationary points.)
3. Solve the inequality $$\frac { x } { x ^ { 2 } - 1 } > 0$$

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

2 Using partial fractions, find \(\int \frac { x } { ( x + 1 ) ( 2 x + 1 ) } \mathrm { d } x\).
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5 Express \(\frac { 4 } { x \left( x ^ { 2 } + 4 \right) }\) in partial fractions.

1. Express \(\frac { 1 } { r ( r + 2 ) }\) in partial fractions.
2. Hence show that \(\sum _ { r = 1 } ^ { n } \frac { 4 } { r ( r + 2 ) } = \frac { n ( 3 n + 5 ) } { ( n + 1 ) ( n + 2 ) }\).

2 Express \(\frac { 6 x ^ { 2 } - 9 x - 16 } { 2 x ^ { 2 } - 5 x - 12 }\) in partial fractions.

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 \(\sec ^ { 2 } x = \mathrm { g } ( y )\).

1 Solve the equation \(\frac { 1 } { x } + \frac { x } { x + 2 } = 1\).

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 \(\int \frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) } \mathrm { d } x\).

7. \(\quad \mathrm { f } ( x ) = \frac { 2 } { x - 1 } - \frac { 6 } { ( x - 1 ) ( 2 x + 1 ) } , x > 1\)

1. Prove that \(\mathrm { f } ( x ) = \frac { 4 } { 2 x + 1 }\).
2. Find the range of f.
3. Find \(\mathrm { f } ^ { - 1 } ( x )\).
4. Find the range of \(\mathrm { f } ^ { - 1 } ( x )\).

1 Express \(\frac { 1 } { r ( r + 1 ) ( r - 1 ) }\) in partial fractions. Find $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$ State the value of $$\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$

11. \(\frac { - 6 x ^ { 2 } + 24 x - 9 } { ( x - 2 ) ( 1 - 3 x ) } \equiv A + \frac { B } { x - 2 } + \frac { C } { 1 - 3 x }\)
a. Find the values of the constants \(A , B\) and \(C\).
b. Using part (a), find \(\mathrm { f } ^ { \prime } ( x )\).
c. Prove that \(\mathrm { f } ( x )\) is an increasing function.

2 Express \(\frac { x + 1 } { ( 2 x - 1 ) }\) in partial fractions.

1. $$f ( x ) = 1 + \frac { 4 x } { 2 x - 5 } - \frac { 15 } { 2 x ^ { 2 } - 7 x + 5 }$$ Show that $$f ( x ) = \frac { 3 x + 2 } { x - 1 }$$

13. $$\frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \equiv A + \frac { B } { ( x - 3 ) } + \frac { C } { ( 1 - 2 x ) }$$ Find the values of the constants \(A , B\) and \(C\).
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7. $$f ( x ) = \frac { 4 x - 5 } { ( 2 x + 1 ) ( x - 3 ) } - \frac { 2 x } { x ^ { 2 } - 9 } , \quad x \neq \pm 3 , x \neq - \frac { 1 } { 2 }$$

1. Show that $$f ( x ) = \frac { 5 } { ( 2 x + 1 ) ( x + 3 ) }$$ The curve \(C\) has equation \(y = \mathrm { f } ( x )\). The point \(P \left( - 1 , - \frac { 5 } { 2 } \right)\) lies on \(C\).
2. Find an equation of the normal to \(C\) at \(P\).

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.

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

7. (i) Show that ( \(2 x + 3\) ) is a factor of ( \(\left. 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)\) and hence, simplify $$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$ (ii) Show that $$\int _ { 2 } ^ { 5 } \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } \mathrm {~d} x = \ln k$$ where \(k\) is an integer.

8 Let \(I = \int \frac { 1 } { x ( 1 + \sqrt { } x ) ^ { 2 } } \mathrm {~d} x\).

1. Show that the substitution \(u = \sqrt { } x\) transforms \(I\) to \(\int \frac { 2 } { u ( 1 + u ) ^ { 2 } } \mathrm {~d} u\).
2. Express \(\frac { 2 } { u ( 1 + u ) ^ { 2 } }\) in the form \(\frac { A } { u } + \frac { B } { 1 + u } + \frac { C } { ( 1 + u ) ^ { 2 } }\).
3. Hence find \(I\).

1. (a) Express \(\frac { 1 } { 4 r ^ { 2 } - 1 }\) in partial fractions.
   (b) Hence prove that

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$ (c) Find the exact value of $$\sum _ { r = 9 } ^ { 25 } \frac { 5 } { 4 r ^ { 2 } - 1 }$$

2 Verify that, for all positive values of \(n\), $$\frac { 1 } { ( n + 2 ) ( 2 n + 3 ) } - \frac { 1 } { ( n + 3 ) ( 2 n + 5 ) } = \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) } .$$ For the series $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ find

1. the sum to \(N\) terms,
2. the sum to infinity.

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 \(x ^ { 2 }\). [5]

2 Express $$\frac { 2 n + 3 } { n ( n + 1 ) }$$ in partial fractions and hence use the method of differences to find $$\sum _ { n = 1 } ^ { N } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$ in terms of \(N\). Deduce the value of $$\sum _ { n = 1 } ^ { \infty } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$

5 Express \(\frac { 12 x ^ { 3 } } { ( 2 x + 1 ) \left( 2 x ^ { 2 } + 1 \right) }\) using partial fractions.