| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration using inverse trig and hyperbolic functions |
| Type | Integration by parts with inverse trig |
| Difficulty | Standard +0.8 Part (a) requires differentiation of inverse tan and algebraic manipulation—straightforward but non-trivial. Part (b) requires integration by parts with inverse trig (choosing u and dv correctly), then applying the result from (a) and evaluating definite integral with exact values—this is a multi-step problem requiring technique mastery beyond routine exercises. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.07l Derivative of ln(x): and related functions1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State correct derivative \(1 - \frac{1}{1+x^2}\) | B1 | |
| Rearrange and obtain the given answer | B1 | AG |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate by parts and reach \(ax^2\tan^{-1}x + b\int\frac{x^2}{1+x^2}dx\) | M1 | |
| Obtain \(\frac{1}{2}x^2\tan^{-1}x - \frac{1}{2}\int\frac{x^2}{1+x^2}dx\), or equivalent | A1 | |
| Obtain complete indefinite integral \(\frac{1}{2}(x^2\tan^{-1}x - x + \tan^{-1}x)\), or equivalent | A1 | |
| Substitute limits having integrated twice | M1 | |
| Obtain the given answer correctly | A1 | AG |
| Total | 5 |
# Question 5:
## Part 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State correct derivative $1 - \frac{1}{1+x^2}$ | B1 | |
| Rearrange and obtain the given answer | B1 | AG |
| **Total** | **2** | |
## Part 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate by parts and reach $ax^2\tan^{-1}x + b\int\frac{x^2}{1+x^2}dx$ | M1 | |
| Obtain $\frac{1}{2}x^2\tan^{-1}x - \frac{1}{2}\int\frac{x^2}{1+x^2}dx$, or equivalent | A1 | |
| Obtain complete indefinite integral $\frac{1}{2}(x^2\tan^{-1}x - x + \tan^{-1}x)$, or equivalent | A1 | |
| Substitute limits having integrated twice | M1 | |
| Obtain the given answer correctly | A1 | AG |
| **Total** | **5** | |
---5 (a) Show that $\frac { \mathrm { d } } { \mathrm { d } x } \left( x - \tan ^ { - 1 } x \right) = \frac { x ^ { 2 } } { 1 + x ^ { 2 } }$.\\
(b) Show that $\int _ { 0 } ^ { \sqrt { 3 } } x \tan ^ { - 1 } x \mathrm {~d} x = \frac { 2 } { 3 } \pi - \frac { 1 } { 2 } \sqrt { 3 }$.\\
\hfill \mbox{\textit{CAIE P3 2020 Q5 [7]}}