Improper algebraic form then partial fractions

7 Let \(\mathrm { f } ( x ) \equiv \frac { x ^ { 2 } + 3 x + 3 } { ( x + 1 ) ( x + 3 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence show that \(\int _ { 0 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x = 3 - \frac { 1 } { 2 } \ln 2\).

10

1. Find the values of the constants \(A , B , C\) and \(D\) such that $$\frac { 2 x ^ { 3 } - 1 } { x ^ { 2 } ( 2 x - 1 ) } \equiv A + \frac { B } { x } + \frac { C } { x ^ { 2 } } + \frac { D } { 2 x - 1 }$$
2. Hence show that $$\int _ { 1 } ^ { 2 } \frac { 2 x ^ { 3 } - 1 } { x ^ { 2 } ( 2 x - 1 ) } \mathrm { d } x = \frac { 3 } { 2 } + \frac { 1 } { 2 } \ln \left( \frac { 16 } { 27 } \right)$$

9 By first expressing \(\frac { 4 x ^ { 2 } + 5 x + 3 } { 2 x ^ { 2 } + 5 x + 2 }\) in partial fractions, show that $$\int _ { 0 } ^ { 4 } \frac { 4 x ^ { 2 } + 5 x + 3 } { 2 x ^ { 2 } + 5 x + 2 } \mathrm {~d} x = 8 - \ln 9$$

8 Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } - 7 x - 1 } { ( x + 1 ) ( 2 x - 3 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that \(\int _ { 2 } ^ { 6 } \mathrm { f } ( x ) \mathrm { d } x = 8 - \ln \left( \frac { 49 } { 3 } \right)\).

7 Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 7 x + 4 } { ( 2 x + 1 ) ( x + 2 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = 8 - \ln 3\).

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

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \ln \left( \frac { 25 } { 8 } \right) - 1\).

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 \(\frac { 9 } { 4 }\).

4. $$f ( x ) = \frac { 2 x ^ { 4 } + 15 x ^ { 3 } + 35 x ^ { 2 } + 21 x - 4 } { ( x + 3 ) ^ { 2 } } \quad x \in \mathbb { R } \quad x > - 3$$

1. Find the values of the constants \(A\), \(B\), \(C\) and \(D\) such that $$\mathrm { f } ( x ) = A x ^ { 2 } + B x + C + \frac { D } { ( x + 3 ) ^ { 2 } }$$
2. Hence find, $$\int \mathrm { f } ( x ) \mathrm { d } x$$

2. $$g ( x ) = \frac { 2 x ^ { 2 } - 5 x + 8 } { x - 2 }$$

1. Write \(g ( x )\) in the form $$A x + B + \frac { C } { x - 2 }$$ where \(A , B\) and \(C\) are integers to be found.
2. Hence use algebraic integration to show that $$\int _ { 4 } ^ { 8 } \mathrm {~g} ( x ) \mathrm { d } x = \alpha + \beta \ln 3$$ where \(\alpha\) and \(\beta\) are integers to be found.

3. Given that $$4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 \equiv ( A x + B ) \left( x ^ { 2 } + 4 \right) + C x + D$$

1. find the values of the constants \(A , B , C\) and \(D\).
2. Hence find $$\int _ { 1 } ^ { 4 } \frac { 4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 } { x ^ { 2 } + 4 } d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.

8. Use partial fractions, and integration, to find the exact value of \(\int _ { 3 } ^ { 4 } \frac { 2 x ^ { 2 } - 3 } { x ( x - 1 ) } \mathrm { d } x\) Write your answer in the form \(a + \ln b\), where \(a\) is an integer and \(b\) is a rational constant.

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 \(k\).

5. a. Given that $$\frac { x ^ { 2 } - 1 } { x + 3 } \equiv x + P + \frac { Q } { x + 3 }$$ find the value of the constant \(P\) and show that \(Q = 8\) \begin{figure}[h] \end{figure} The curve \(C\) has equation \(y = \mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = \frac { x ^ { 2 } - 1 } { x + 3 } \quad x > - 3$$ Figure 3 shows a sketch of the curve \(C\).
The region \(R\), shown shaded in Figure 4, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = 5\).
b. Find the exact area of \(R\), writing your answer in the form \(a \ln 2\), where \(a\) is constant to be found.
(4)

1. (a) Given that

$$\frac { x ^ { 2 } + 8 x - 3 } { x + 2 } \equiv A x + B + \frac { C } { x + 2 } \quad x \in \mathbb { R } \quad x \neq - 2$$ find the values of the constants \(A , B\) and \(C\) (b) Hence, using algebraic integration, find the exact value of $$\int _ { 0 } ^ { 6 } \frac { x ^ { 2 } + 8 x - 3 } { x + 2 } d x$$ giving your answer in the form \(a + b \ln 2\) where \(a\) and \(b\) are integers to be found.

\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = f(x)\), where $$f(x) = \frac{x^2 + 1}{(1 + x)(3 - x)}, \quad 0 \leq x < 3.$$

1. Given that \(f(x) = A + \frac{B}{1 + x} + \frac{C}{3 - x}\), find the values of the constants \(A\), \(B\) and \(C\). [4]

The finite region \(R\), shown in Fig. 1, is bounded by the curve with equation \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).

1. Find the area of \(R\), giving your answer in the form \(p + q \ln r\), where \(p\), \(q\) and \(r\) are rational constants to be found. [5]