Partial fractions with linear factors – decompose, integrate, and expand as series

Express a proper rational function with distinct linear factors in partial fractions, evaluate a definite integral, and also find a binomial/power series expansion.

4 questions · Standard +0.3

1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<11.08j Integration using partial fractions
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OCR C4 Q9
13 marks Standard +0.3
9. $$f ( x ) = \frac { 8 - x } { ( 1 + x ) ( 2 - x ) } , \quad | x | < 1$$
  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.
OCR C4 Q6
12 marks Standard +0.3
6. $$f ( x ) = \frac { 1 + 3 x } { ( 1 - x ) ( 1 - 3 x ) } , \quad | x | < \frac { 1 } { 3 }$$
  1. Find the values of the constants \(A\) and \(B\) such that $$\mathrm { f } ( x ) = \frac { A } { 1 - x } + \frac { B } { 1 - 3 x }$$
  2. Evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } } f ( x ) d x$$ giving your answer as a single logarithm.
  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.
Edexcel C4 Q7
14 marks Standard +0.3
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
Edexcel C4 Q3
13 marks Standard +0.3
$$f(x) = \frac{1 + 14x}{(1 - x)(1 + 2x)}, \quad |x| < \frac{1}{2}.$$
  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]