1.08j Integration using partial fractions

8

1. Evaluate exactly \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { - x } \mathrm {~d} x\).
2. Find \(\int \frac { x - 1 } { x + 1 } \mathrm {~d} x\).

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 \(\int _ { 2 } ^ { 5 } \frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) } \mathrm { d } x = \ln 24\).

8

1. Express \(\frac { 7 x ^ { 2 } - 12 x + 1 } { \left( x ^ { 2 } + 1 \right) ( x - 2 ) }\) in the form \(\frac { A x + B } { x ^ { 2 } + 1 } + \frac { C } { x - 2 }\) where \(A , B\) and \(C\) are constants to be found.
2. Hence find the exact value of \(\int _ { 0 } ^ { 1 } \frac { 7 x ^ { 2 } - 12 x + 1 } { \left( x ^ { 2 } + 1 \right) ( x - 2 ) } \mathrm { d } x\).

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 \(\int _ { 2 } ^ { 5 } \frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) } \mathrm { d } x = \ln 24\).

a) Express \(\frac { 9 } { ( x - 1 ) ( x + 2 ) ^ { 2 } }\) in terms of partial fractions. b) Find \(\int \frac { 9 } { ( x - 1 ) ( x + 2 ) ^ { 2 } } \mathrm {~d} x\).

The polynomial \(p(x)\) is defined by \(p(x) = 9x^3 + 18x^2 + 5x + 4\).

1. Find the quotient when \(p(x)\) is divided by \((3x + 2)\), and show that the remainder is 6. [3]
2. Find the value of \(\int_0^2 \frac{p(x)}{3x + 2} \, dx\), giving your answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers. [5]

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]

Let \(f(x) = \frac{7x^2 - 15x + 8}{(1 - 2x)(2 - x)^2}\).

1. Express \(f(x)\) in partial fractions. [5]
2. Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]

Let \(f(x) = \frac{6x^2 + 8x + 9}{(2 - x)(3 + 2x)^2}\).

1. Express \(f(x)\) in partial fractions. [5]
2. Hence, showing all necessary working, show that \(\int_{-1}^0 f(x) dx = 1 + \frac{1}{2}\ln\left(\frac{4}{3}\right)\). [5]

$$f(x) = \frac{1 + 14x}{(1 - x)(1 + 2x)}, \quad |x| < \frac{1}{2}.$$

1. Express \(f(x)\) in partial fractions. [3]
2. Hence find the exact value of \(\int_{-\frac{1}{6}}^{\frac{1}{4}} f(x) \, dx\), giving your answer in the form \(\ln p\), where \(p\) is rational. [5]
3. Use the binomial theorem to expand \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^5\), simplifying each term. [5]

\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]

$$\text{f}(x) = \frac{25}{(3 + 2x)^2(1 - x)}, \quad |x| < 1.$$

1. Express f(x) as a sum of partial fractions. [4]
2. Hence find \(\int \text{f}(x) \, dx\). [5]
3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^2\). Give each coefficient as a simplified fraction. [7]

\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]

Using partial fractions, find \(\int \frac{x}{(x+1)(2x+1)} \, dx\). [7]

$$\text{f}(x) = \frac{x(3x-7)}{(1-x)(1-3x)}, \quad |x| < \frac{1}{3}.$$

1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$\text{f}(x) = A + \frac{B}{1-x} + \frac{C}{1-3x}.$$ [4]
2. Evaluate $$\int_0^{\frac{1}{4}} \text{f}(x) \, dx,$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational. [5]
3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]