| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Sequential multi-part (building on previous) |
| Difficulty | Standard +0.3 Part (i) is a standard integration by parts application with a straightforward choice of u and dv. Part (ii) requires the additional insight to use the identity tan²x = sec²x - 1, then apply the result from (i), making it slightly above average difficulty but still a routine C4 question testing standard techniques. |
| Spec | 1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Parts using correct split of \(u=x\), \(\frac{dv}{dx}=\sec^2 x\) | M1 | 1st stage result of form \(f(x)+/-\int g(x)dx\) |
| \(x\tan x - \int \tan x\, dx\) | A1 | Correct 1st stage |
| \(\int \tan x\, dx = -\ln\cos x\) or \(\ln\sec x\) | B1 | |
| \(x\tan x + \ln\cos x + c\) or \(x\tan x - \ln\sec x + c\) | A1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\tan^2 x = +/-\sec^2 x +/- 1\) | M1 | or \(\sec^2 x = +/-1+/-\tan^2 x\) |
| \(\int x\sec^2 x\, dx - \int x\, dx\) s.o.i. | A1 | Correct 1st stage |
| \(x\tan x + \ln\cos x - \frac{1}{2}x^2 + c\) | A1\(\sqrt{}\) | 3 f.t. their answer to part (i) \(-\frac{1}{2}x^2\) |
# Question 4:
## Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Parts using correct split of $u=x$, $\frac{dv}{dx}=\sec^2 x$ | M1 | 1st stage result of form $f(x)+/-\int g(x)dx$ |
| $x\tan x - \int \tan x\, dx$ | A1 | Correct 1st stage |
| $\int \tan x\, dx = -\ln\cos x$ or $\ln\sec x$ | B1 | |
| $x\tan x + \ln\cos x + c$ or $x\tan x - \ln\sec x + c$ | A1 | **4** |
## Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\tan^2 x = +/-\sec^2 x +/- 1$ | M1 | or $\sec^2 x = +/-1+/-\tan^2 x$ |
| $\int x\sec^2 x\, dx - \int x\, dx$ s.o.i. | A1 | Correct 1st stage |
| $x\tan x + \ln\cos x - \frac{1}{2}x^2 + c$ | A1$\sqrt{}$ | **3** f.t. their answer to part (i) $-\frac{1}{2}x^2$ |4 (i) Use integration by parts to find $\int x \sec ^ { 2 } x \mathrm {~d} x$.\\
(ii) Hence find $\int x \tan ^ { 2 } x \mathrm {~d} x$.
\hfill \mbox{\textit{OCR C4 2006 Q4 [7]}}