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 · Moderate -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 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 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 \([ 0,0.5 ]\)
Edexcel FP3 Q22
7 marks Standard +0.8
$$4x^2 + 4x + 17 \equiv (ax + b)^2 + c, \quad a > 0.$$
  1. Find the values of \(a\), \(b\) and \(c\). [3]
  2. Find the exact value of $$\int_{-0.5}^{1.5} \frac{1}{4x^2 + 4x + 17} \, dx.$$ [4]
OCR FP2 2012 January Q2
5 marks Easy -2.5
By first completing the square in the denominator, find the exact value of $$\int_{\frac{1}{2}}^{\frac{1}{2}} \frac{1}{4x^2 - 4x + 5} dx.$$ [5]