Partial fractions with linear factors – decompose and integrate (definite)

5

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

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

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.

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. The curve \(C\) with equation

$$y = \frac { p - 3 x } { ( 2 x - q ) ( x + 3 ) } \quad x \in \mathbb { R } , x \neq - 3 , x \neq 2$$ where \(p\) and \(q\) are constants, passes through the point \(\left( 3 , \frac { 1 } { 2 } \right)\) and has two vertical asymptotes
with equations \(x = 2\) and \(x = - 3\) with equations \(x = 2\) and \(x = - 3\)

1. 1. Explain why you can deduce that \(q = 4\)
   2. Show that \(p = 15\) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-38_616_889_842_587} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure} Figure 4 shows a sketch of part of the curve \(C\). The region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 3\)
2. Show that the exact value of the area of \(R\) is \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are rational constants to be found.

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]

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.

8

1. Given that \(\frac { 2 x + 11 } { ( 2 x + 1 ) ( x + 3 ) } \equiv \frac { A } { 2 x + 1 } + \frac { B } { x + 3 }\), find the values of the constants \(A\) and \(B\).
2. Hence show that \(\int _ { 0 } ^ { 2 } \frac { 2 x + 11 } { ( 2 x + 1 ) ( x + 3 ) } \mathrm { d } x = \ln 15\).

7

1. Express \(\frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { x + 1 }\) where \(A\) and \(B\) are constants.
2. Hence show that \(\int _ { 2 } ^ { 5 } \frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) } \mathrm { d } x = \ln 24\).

7

1. Express \(\frac { 8 x - 1 } { ( 2 x - 1 ) ( x - 1 ) }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { x + 1 }\) where \(A\) and \(B\) are constants.
2. Hence show that \(\equiv \frac { 5 x - 1 } { \overline { 2 } } \frac { 8 x - 1 ) ( x + 1 ) } { ( 2 x - \ln 24 \text {. } }\)

7

1. Express \(\frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { x + 1 }\) where \(A\) and \(B\) are constants.
2. Hence show that \(\int _ { 2 } ^ { 5 } \frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) } \mathrm { d } x = \ln 24\).

Let \(\text{f}(x) = \frac{5a}{(2x - a)(3a - x)}\), where \(a\) is a positive constant.

1. Express f\((x)\) in partial fractions. [3]
2. Hence show that \(\int_a^{2a} \text{f}(x) \, dx = \ln 6\). [4]

The function \(f\) is given by $$f(x) = \frac{25x + 32}{(2x - 5)(x + 1)(x + 2)}.$$

1. Express \(f(x)\) in terms of partial fractions. [4]
2. Show that \(\int_1^2 f(x) dx = -\ln P\), where \(P\) is an integer whose value is to be found. [5]
3. Show that the sign of \(f(x)\) changes in the interval \(x = 2\) to \(x = 3\). Explain why the change of sign method fails to locate a root of the equation \(f(x) = 0\) in this case. [2]