Question 2 - A-Level Maths

Edexcel C4 2008 June — Question 2 6 marks

Exam Board	Edexcel
Module	C4 (Core Mathematics 4)
Year	2008
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Integration by Parts
Type	Sequential multi-part (building on previous)
Difficulty	Moderate -0.3 Part (a) is a textbook integration by parts example with standard choices u=x, dv=e^x dx. Part (b) requires applying the technique again using the result from (a), making it slightly more involved but still a routine two-step application of the same method. This is standard C4 fare with no conceptual surprises, placing it slightly below average difficulty.
Spec	1.08i Integration by parts

2. (a) Use integration by parts to find \(\int x \mathrm { e } ^ { x } \mathrm {~d} x\).
(b) Hence find \(\int x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x\).

Show mark scheme Show mark scheme source

Question 2:

Part (a)

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
\(u = x \Rightarrow \frac{du}{dx} = 1\); \(\frac{dv}{dx} = e^x \Rightarrow v = e^x\)
\(\int xe^x\,dx = xe^x - \int e^x \cdot 1\,dx\)	M1	Use of integration by parts formula in correct direction
Correct expression (ignore \(dx\))	A1
\(= xe^x - e^x (+c)\)	A1	Correct integration with/without \(+c\)

Part (b)

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
\(u = x^2 \Rightarrow \frac{du}{dx} = 2x\); \(\frac{dv}{dx} = e^x \Rightarrow v = e^x\)
\(\int x^2 e^x\,dx = x^2 e^x - \int e^x \cdot 2x\,dx\)	M1	Use of integration by parts in correct direction
Correct expression (ignore \(dx\))	A1
\(= x^2 e^x - 2(xe^x - e^x) + c\)	A1 ISW	Correct expression including \(+c\), seen at any stage in part (b)

# Question 2:

## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $u = x \Rightarrow \frac{du}{dx} = 1$; $\frac{dv}{dx} = e^x \Rightarrow v = e^x$ | | |
| $\int xe^x\,dx = xe^x - \int e^x \cdot 1\,dx$ | M1 | Use of integration by parts formula in correct direction |
| Correct expression (ignore $dx$) | A1 | |
| $= xe^x - e^x (+c)$ | A1 | Correct integration with/without $+c$ |

## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $u = x^2 \Rightarrow \frac{du}{dx} = 2x$; $\frac{dv}{dx} = e^x \Rightarrow v = e^x$ | | |
| $\int x^2 e^x\,dx = x^2 e^x - \int e^x \cdot 2x\,dx$ | M1 | Use of integration by parts in correct direction |
| Correct expression (ignore $dx$) | A1 | |
| $= x^2 e^x - 2(xe^x - e^x) + c$ | A1 ISW | Correct expression including $+c$, seen at any stage in part (b) |

---

Show LaTeX source

2. (a) Use integration by parts to find $\int x \mathrm { e } ^ { x } \mathrm {~d} x$.\\
(b) Hence find $\int x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x$.\\

\hfill \mbox{\textit{Edexcel C4 2008 Q2 [6]}}

This paper (8 questions)

View full paper

Q1 4 Q2 6 Q3 8 Q4 9 Q5 9 Q6 12 Q7 11 Q8 16