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.

6 questions · Challenging +1.0

1.08i Integration by parts
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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.
Edexcel FP3 Q15
13 marks Standard +0.8
$$I_n = \int_0^1 x^n e^x \, dx \text{ and } J_n = \int_0^1 x^n e^{-x} \, dx, \quad n \geq 0.$$
  1. Show that, for \(n \geq 1\), $$I_n = e - nI_{n-1}.$$ [2]
  2. Find a similar reduction formula for \(J_n\). [3]
  3. Show that \(J_2 = 2 - \frac{5}{e}\). [3]
  4. Show that \(\int_0^1 x^n \cosh x \, dx = \frac{1}{2}(I_n + J_n)\). [1]
  5. Hence, or otherwise, evaluate \(\int_0^1 x^2 \cosh x \, dx\), giving your answer in terms of \(e\). [4]
Edexcel FP3 Q32
8 marks Challenging +1.2
$$I_n = \int_0^1 x^n e^{2x} \, dx, \quad n \geq 0.$$
  1. Prove that, for \(n \geq 1\), $$I_n = \frac{1}{2}(x^n e^{2x} - nI_{n-1}).$$ [3]
  2. Find, in terms of \(e\), the exact value of $$\int_0^1 x^2 e^{2x} \, dx.$$ [5]