Edexcel C4 — Question 2 7 marks

Exam BoardEdexcel
ModuleC4 (Core Mathematics 4)
Marks7
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Mark schemeDownload PDF ↗
TopicIntegration by Parts
TypeDouble integration by parts
DifficultyStandard +0.3 This is a standard double integration by parts question requiring two applications of the technique with a polynomial and exponential. While it requires careful execution and algebraic manipulation, it follows a well-practiced procedure with no conceptual surprises—slightly easier than average for C4 level.
Spec1.08i Integration by parts
2. Use integration by parts to find $$\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$
AnswerMarks Guidance
\(u = x^2, u' = 2x, v' = e^x, v = -e^{-x}\)M1 A1
\(I = -x^2 e^{-x} - \int -2x e^{-x} dx = -x^2 e^{-x} + \int 2x e^{-x} dx\)A2
\(u = 2x, u' = 2, v' = e^{-x}, v = -e^{-x}\)M1
\(I = -x^2 e^{-x} - 2x e^{-x} - \int -2e^{-x} dx\)A1
\(= -x^2 e^{-x} - 2x e^{-x} - 2e^{-x} + c\)A1 (7 marks) 
$u = x^2, u' = 2x, v' = e^x, v = -e^{-x}$ | M1 A1 |
$I = -x^2 e^{-x} - \int -2x e^{-x} dx = -x^2 e^{-x} + \int 2x e^{-x} dx$ | A2 |

$u = 2x, u' = 2, v' = e^{-x}, v = -e^{-x}$ | M1 |
$I = -x^2 e^{-x} - 2x e^{-x} - \int -2e^{-x} dx$ | A1 |

$= -x^2 e^{-x} - 2x e^{-x} - 2e^{-x} + c$ | A1 | (7 marks)
2. Use integration by parts to find

$$\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$

\hfill \mbox{\textit{Edexcel C4  Q2 [7]}}
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