Improper integrals with infinite upper limit (exponential/IBP)

Evaluate an improper integral with an infinite upper limit where the integrand involves exponential functions, typically requiring integration by parts and explicit limiting process (e.g., showing x·e^(-kx) → 0 as x → ∞).

5 questions · Standard +0.7

4.08c Improper integrals: infinite limits or discontinuous integrands
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AQA FP3 2008 January Q4
7 marks Standard +0.3
4
  1. Explain why \(\int _ { 1 } ^ { \infty } x \mathrm { e } ^ { - 3 x } \mathrm {~d} x\) is an improper integral.
  2. Find \(\int x \mathrm { e } ^ { - 3 x } \mathrm {~d} x\).
  3. Hence evaluate \(\int _ { 1 } ^ { \infty } x \mathrm { e } ^ { - 3 x } \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2012 January Q5
8 marks Challenging +1.3
5
  1. Explain why \(\int _ { \frac { 1 } { 2 } } ^ { \infty } \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x\) is an improper integral.
    (1 mark)
  2. By using the substitution \(u = x ^ { 2 } \mathrm { e } ^ { - 4 x } + 3\), find $$\int \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x$$
  3. Hence evaluate \(\int _ { \frac { 1 } { 2 } } ^ { \infty } \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2010 June Q3
7 marks Standard +0.3
3
  1. Explain why \(\int _ { 1 } ^ { \infty } 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x\) is an improper integral.
  2. Find \(\quad \int 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x\).
  3. Hence evaluate \(\int _ { 1 } ^ { \infty } 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2015 June Q4
7 marks Standard +0.8
4
  1. Explain why \(\int _ { 2 } ^ { \infty } ( x - 2 ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x\) is an improper integral.
  2. Evaluate \(\int _ { 2 } ^ { \infty } ( x - 2 ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), showing the limiting process used.
AQA Further Paper 2 2019 June Q13
10 marks Standard +0.8
  1. Explain why \(\int_3^{\infty} x^2 e^{-2x} \, dx\) is an improper integral. [1 mark]
  2. Evaluate \(\int_3^{\infty} x^2 e^{-2x} \, dx\) Show the limiting process. [9 marks]