| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Show definite integral equals specific value (algebraic/exponential substitution) |
| Difficulty | Challenging +1.2 This is a structured integration question requiring substitution u = tan x (a standard technique), followed by algebraic manipulation and pattern recognition. Part (i) guides students through the key result, and part (ii) applies it systematically. While requiring multiple steps and careful algebra, the substitution is given, the approach is clearly signposted, and the techniques are standard for P3 level—making this moderately above average but not requiring novel insight. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(\frac{du}{dx} = \sec^2 x\) | B1 | |
| Express integrand in terms of \(u\) and \(du\) | M1 | |
| Integrate to obtain \(\frac{u^{n+1}}{n+1}\) or equivalent | A1 | |
| Substitute correct limits to confirm \(\frac{1}{n+1}\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use \(\sec^2 x = 1 + \tan^2 x\) twice | M1 | |
| Obtain integrand \(\tan^4 x + \tan^2 x\) | A1 | |
| Apply result from part (i) to obtain \(\frac{1}{3}\) | A1 | [3] |
| Or: Use \(\sec^2 x = 1 + \tan^2 x\) and substitution from (i) | M1 | |
| Obtain \(\int u^2\, du\) | A1 | |
| Apply limits correctly and obtain \(\frac{1}{3}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Arrange integrand to \(t^9 + t^7 + 4(t^7 + t^5) + t^5 + t^3\) | B1 | |
| Attempt application of result from part (i) at least twice | M1 | |
| Obtain \(\frac{1}{8} + \frac{4}{6} + \frac{1}{4}\) and hence \(\frac{25}{24}\) or exact equivalent | A1 | [3] |
## Question 10:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $\frac{du}{dx} = \sec^2 x$ | B1 | |
| Express integrand in terms of $u$ and $du$ | M1 | |
| Integrate to obtain $\frac{u^{n+1}}{n+1}$ or equivalent | A1 | |
| Substitute correct limits to confirm $\frac{1}{n+1}$ | A1 | [4] |
### Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use $\sec^2 x = 1 + \tan^2 x$ twice | M1 | |
| Obtain integrand $\tan^4 x + \tan^2 x$ | A1 | |
| Apply result from part (i) to obtain $\frac{1}{3}$ | A1 | [3] |
| **Or:** Use $\sec^2 x = 1 + \tan^2 x$ and substitution from (i) | M1 | |
| Obtain $\int u^2\, du$ | A1 | |
| Apply limits correctly and obtain $\frac{1}{3}$ | A1 | |
### Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Arrange integrand to $t^9 + t^7 + 4(t^7 + t^5) + t^5 + t^3$ | B1 | |
| Attempt application of result from part (i) at least twice | M1 | |
| Obtain $\frac{1}{8} + \frac{4}{6} + \frac{1}{4}$ and hence $\frac{25}{24}$ or exact equivalent | A1 | [3] |10 (i) Use the substitution $u = \tan x$ to show that, for $n \neq - 1$,
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { n + 2 } x + \tan ^ { n } x \right) \mathrm { d } x = \frac { 1 } { n + 1 }$$
(ii) Hence find the exact value of
\begin{enumerate}[label=(\alph*)]
\item $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sec ^ { 4 } x - \sec ^ { 2 } x \right) \mathrm { d } x$,
\item $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 9 } x + 5 \tan ^ { 7 } x + 5 \tan ^ { 5 } x + \tan ^ { 3 } x \right) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2011 Q10 [10]}}