Exponential times polynomial

A question is this type if and only if I_n involves e^(ax) or e^(f(x)) multiplied by x^n or a polynomial in x, where the exponential is the primary function being integrated with a polynomial factor.

4 questions · Challenging +1.0

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OCR FP2 2013 January Q4
8 marks Standard +0.8
4 You are given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { 2 x } \mathrm {~d} x\) for \(n \geqslant 0\).
  1. Show that \(I _ { n } = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } - \frac { 1 } { 2 } n I _ { n - 1 }\) for \(n \geqslant 1\).
  2. Find \(I _ { 3 }\) in terms of e.
OCR FP2 2015 June Q4
9 marks Challenging +1.2
4 It is given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x } \mathrm {~d} x\) for \(n \geqslant 0\).
  1. Show that \(I _ { n } = n I _ { n - 1 } + k\) for \(n \geqslant 1\), where \(k\) is a constant to be determined.
  2. Find the exact value of \(I _ { 3 }\).
  3. Find the exact value of \(990 I _ { 8 } - I _ { 11 }\).
Edexcel FP2 2023 June Q8
7 marks Challenging +1.2
8. $$I _ { n } = \int _ { 0 } ^ { 2 } ( x - 2 ) ^ { n } \mathrm { e } ^ { 4 x } \mathrm {~d} x \quad n \geqslant 0$$
  1. Prove that for \(n \geqslant 1\) $$I _ { n } = - a ^ { n - 2 } - \frac { n } { 4 } I _ { n - 1 }$$ where \(a\) is a constant to be determined.
  2. Hence determine the exact value of $$\int _ { 0 } ^ { 2 } ( x - 2 ) ^ { 2 } e ^ { 4 x } d x$$
OCR FP2 Q6
8 marks Standard +0.8
6
  1. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { - x } x ^ { n } \mathrm {~d} x$$ Prove that, for \(n \geqslant 1\), $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
  2. Evaluate \(I _ { 3 }\), giving the answer in terms of e.