| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Derivative then integrate by parts |
| Difficulty | Standard +0.3 This is a guided integration by parts question where part (i) provides the key derivative needed for part (ii). Once students recognize that d/dx(e^(cos x)) = -sin x ยท e^(cos x), the integral becomes straightforward using integration by parts with u = cos x. The structure is more routine than average since the derivative is given explicitly, requiring only pattern recognition and standard technique application. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.08i Integration by parts |
| Answer | Marks |
|---|---|
| \(-\sin x\, e^{\cos x}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int \sin x\, e^{\cos x}\,dx = -e^{\cos x}\) | B1 | anywhere in part (ii) |
| Parts with split \(u = \cos x,\, dv = \sin x\, e^{\cos x}\) | M1 | result \(f(x) +/- \int g(x)\,dx\) |
| Indef. Integ, 1st stage \(-\cos x\, e^{\cos x} - \int \sin x\, e^{\cos x}\,dx\) | A1 | accept \(... -\int -e^{\cos x} \cdot -\sin x\,dx\) |
| Second stage \(= -\cos x\, e^{\cos x} + e^{\cos x}\) | *A1 | |
| Final answer \(= 1\) | dep*A2 |
# Question 8:
## Part (i):
| $-\sin x\, e^{\cos x}$ | B1 | |
|---|---|---|
## Part (ii):
| $\int \sin x\, e^{\cos x}\,dx = -e^{\cos x}$ | B1 | anywhere in part (ii) |
|---|---|---|
| Parts with split $u = \cos x,\, dv = \sin x\, e^{\cos x}$ | M1 | result $f(x) +/- \int g(x)\,dx$ |
| Indef. Integ, 1st stage $-\cos x\, e^{\cos x} - \int \sin x\, e^{\cos x}\,dx$ | A1 | accept $... -\int -e^{\cos x} \cdot -\sin x\,dx$ |
| Second stage $= -\cos x\, e^{\cos x} + e^{\cos x}$ | *A1 | |
| Final answer $= 1$ | dep*A2 | |
---8 (i) State the derivative of $\mathrm { e } ^ { \cos x }$.\\
(ii) Hence use integration by parts to find the exact value of
$$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos x \sin x \mathrm { e } ^ { \cos x } \mathrm {~d} x$$
\hfill \mbox{\textit{OCR C4 2010 Q8 [7]}}