Standard integral of 1/(a²+x²)

A question is this type if and only if it requires direct application of the standard result that ∫1/(a²+x²)dx = (1/a)arctan(x/a) + c, possibly after completing the square.

5 questions · Standard +0.3

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Edexcel F3 2024 January Q1
7 marks Moderate -0.3
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Show that $$\int _ { 4 } ^ { 4 \sqrt { 3 } } \frac { 8 } { 16 + x ^ { 2 } } d x = p \pi$$ where \(p\) is a rational number to be determined.
  2. Determine the exact value of \(k\) for which $$\int _ { \frac { 3 } { 4 } } ^ { k } \frac { 2 } { \sqrt { 9 - 4 x ^ { 2 } } } d x = \frac { \pi } { 12 }$$
Edexcel F3 2021 June Q7
8 marks Standard +0.8
  1. Using calculus, find the exact values of
    1. \(\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } - 4 x + 5 } \mathrm {~d} x\)
    2. \(\int _ { \sqrt { 3 } } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 3 } } { x ^ { 2 } } \mathrm {~d} x\)
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Edexcel FP3 2010 June Q2
5 marks Standard +0.3
2. Use calculus to find the exact value of \(\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x\).
OCR FP2 2012 January Q2
5 marks Standard +0.3
2 By first completing the square in the denominator, find the exact value of $$\int _ { \frac { 1 } { 2 } } ^ { \frac { 3 } { 2 } } \frac { 1 } { 4 x ^ { 2 } - 4 x + 5 } \mathrm {~d} x$$
OCR Further Pure Core 1 2018 September Q4
6 marks Standard +0.3
4 In this question you must show detailed reasoning.
Find the exact value of each of the following.
  1. \(\int _ { 1 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 2 x + 10 } \mathrm {~d} x\)
  2. The mean value of \(\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\) in the interval \([ 0,0.5 ]\)