| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Topic | Integration by Parts |
| Type | Integration of e^(ax)·trig(bx) |
| Difficulty | Standard +0.3 This is a standard integration by parts question with a well-known technique (applying it twice to get back to the original integral). Part (i) is a guided 'show that' proof requiring two applications of a familiar method, and part (ii) is straightforward substitution to find a particular solution. While it requires careful algebraic manipulation, it's a textbook exercise with no novel insight needed, making it slightly easier than average. |
| Spec | 1.08i Integration by parts |
**Question 8(i) and 8(ii)**
**(i)**
- $e^x\sin x - \int e^x\cos x\,dx$ or $-e^x\cos x + \int e^x\cos x\,dx$ **M1** Attempt integration by parts once
- **A1** Obtain correct expression
- $e^x\sin x - \left(e^x\cos x + \int e^x\sin x\,dx\right)$ or $-e^x\cos x + \left(e^x\sin x - \int e^x\sin x\,dx\right)$ **M1** Attempt integration by parts again on their integral (parts consistent with first stage)
- **A1** Obtain correct expression – must be entire expression, including '$uv$' from first stage
- $2\int e^x\sin x\,dx = e^x\sin x - e^x\cos x$ **M1** Attempt to rearrange
- $\int e^x\sin x\,dx = \frac{1}{2}e^x(\sin x - \cos x) + c$ **A1** Obtain correct integral **A.G.**
**(ii)**
- $2 = \frac{1}{2}(0-1) + c$ so $c = \frac{5}{2}$ **M1** Attempt to find $c$, using $x=0$ and $y=2$
- $y = \frac{1}{2}e^x(\sin x - \cos x) + \frac{5}{2}$ **A1** Obtain correct equation aef8 (i) Use integration by parts twice to show that
$$\int \mathrm { e } ^ { x } \sin x \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { x } ( \sin x - \cos x ) + c .$$
(ii) Hence find the equation of the curve which passes through the point $( 0,2 )$ and for which $\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { x } \sin x$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2018 Q8 [8]}}