Improper integral evaluation

Evaluate an improper integral with infinite limits or discontinuous integrand, or explain why it doesn't converge.

11 questions · Moderate -0.1

4.08c Improper integrals: infinite limits or discontinuous integrands
Sort by: Default | Easiest first | Hardest first
OCR C2 2006 January Q6
8 marks Standard +0.3
6
  1. Find \(\int \left( x ^ { \frac { 1 } { 2 } } + 4 \right) \mathrm { d } x\).
    1. Find the value, in terms of \(a\), of \(\int _ { 1 } ^ { a } 4 x ^ { - 2 } \mathrm {~d} x\), where \(a\) is a constant greater than 1 .
    2. Deduce the value of \(\int _ { 1 } ^ { \infty } 4 x ^ { - 2 } \mathrm {~d} x\).
OCR C2 Q6
10 marks Standard +0.3
6. Evaluate
  1. \(\quad \int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 5 x + 4 \right) \mathrm { d } x\),
  2. \(\int _ { - \infty } ^ { - 1 } \frac { 1 } { x ^ { 4 } } \mathrm {~d} x\).
OCR C2 2011 January Q6
8 marks Moderate -0.3
6
  1. Find \(\int \frac { x ^ { 3 } + 3 x ^ { \frac { 1 } { 2 } } } { x } \mathrm {~d} x\).
    1. Find, in terms of \(a\), the value of \(\int _ { 2 } ^ { a } 6 x ^ { - 4 } \mathrm {~d} x\), where \(a\) is a constant greater than 2 .
    2. Deduce the value of \(\int _ { 2 } ^ { \infty } 6 x ^ { - 4 } \mathrm {~d} x\).
OCR C2 2013 June Q4
8 marks Standard +0.3
4
  1. Find \(\int \left( 5 x ^ { 3 } - 6 x + 1 \right) \mathrm { d } x\).
    1. Find \(\int 24 x ^ { - 3 } \mathrm {~d} x\).
    2. Given that \(\int _ { a } ^ { \infty } 24 x ^ { - 3 } \mathrm {~d} x = 3\), find the value of the positive constant \(a\).
OCR C2 2016 June Q5
8 marks Standard +0.3
5
  1. Find \(\int \left( x ^ { 2 } + 2 \right) ( 2 x - 3 ) \mathrm { d } x\).
    1. Find, in terms of \(a\), the value of \(\int _ { 1 } ^ { a } \left( 6 x ^ { - 2 } - 4 x ^ { - 3 } \right) \mathrm { d } x\), where \(a\) is a constant greater than 1 .
    2. Deduce the value of \(\int _ { 1 } ^ { \infty } \left( 6 x ^ { - 2 } - 4 x ^ { - 3 } \right) \mathrm { d } x\).
OCR C2 Q8
11 marks Standard +0.3
  1. (i) Given that
$$\int _ { 1 } ^ { 3 } \left( x ^ { 2 } - 2 x + k \right) d x = 8 \frac { 2 } { 3 }$$ find the value of the constant \(k\).
(ii) Evaluate $$\int _ { 2 } ^ { \infty } \frac { 6 } { x ^ { \frac { 5 } { 2 } } } \mathrm {~d} x$$ giving your answer in its simplest form.
AQA Further Paper 1 2020 June Q1
1 marks Easy -1.8
1 Which of the integrals below is not an improper integral?
Circle your answer. \(\int _ { 0 } ^ { \infty } e ^ { - x } d x\) \(\int _ { 0 } ^ { 2 } \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x\) \(\int _ { 0 } ^ { 1 } \sqrt { x } \mathrm {~d} x\) \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\)
Edexcel C2 Q1
4 marks Moderate -0.3
Evaluate \(\int_0^1 \frac{1}{\sqrt{x}} \, dx\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers. [4]
Edexcel CP1 2021 June Q5
7 marks Moderate -0.3
  1. Evaluate the improper integral $$\int_1^{\infty} 2e^{-\frac{1}{2}x} dx$$ [3]
  2. The air temperature, \(\theta ^{\circ}C\), on a particular day in London is modelled by the equation $$\theta = 8 - 5\sin\left(\frac{\pi}{12}t\right) - \cos\left(\frac{\pi}{6}t\right) \quad 0 \leq t \leq 24$$ where \(t\) is the number of hours after midnight.
    1. Use calculus to show that the mean air temperature on this day is \(8^{\circ}C\), according to the model. [3] Given that the actual mean air temperature recorded on this day was higher than \(8^{\circ}C\),
    2. explain how the model could be refined. [1]
SPS SPS FM 2020 December Q2
4 marks Moderate -0.3
Let \(a, b\) satisfy \(0 < a < b\).
  1. Find, in terms of \(a\) and \(b\), the value of $$\int_a^b \frac{81}{x^4} dx$$ [2]
  2. Explaining clearly any limiting processes used, find the value of \(a\), given that $$\int_a^{\infty} \frac{81}{x^4} dx = \frac{216}{125}$$ [2]
OCR Further Pure Core 2 2021 June Q2
5 marks Standard +0.3
In this question you must show detailed reasoning. Show that \(\int_5^{\infty} (x-1)^{-2} dx = 1\). [5]