| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Show integral transforms via substitution then evaluate (algebraic/exponential) |
| Difficulty | Standard +0.3 This is a standard integration by substitution question with a given substitution and target form. Part (i) requires routine algebraic manipulation after substitution (changing limits, expressing x^5 in terms of u, and simplifying). Part (ii) involves expanding and integrating a straightforward rational function. While it requires careful algebra across multiple steps, it follows a predictable template with no novel insight needed, making it slightly easier than average. |
| Spec | 1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(du = 2x \, dx\), or equivalent | B1 | |
| Substitute for \(x\) and \(dx\) throughout | M1 | |
| Reduce to the given form and justify the change in limits | A1 | [3] |
| (ii) Convert integrand to a sum of integrable terms and attempt integration | M1 | |
| Obtain integral \(\frac{1}{2}\ln u + \frac{1}{u} - \frac{1}{4u^2}\), or equivalent | A1 + A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Substitute limits in an integral containing two terms of the form \(a \ln u\) and \(bu^{-2}\) | M1 | |
| Obtain answer \(\frac{1}{2}\ln 2 - \frac{7}{16}\), exact simplified equivalent | A1 | [5] |
**(i)** State or imply $du = 2x \, dx$, or equivalent | B1 |
Substitute for $x$ and $dx$ throughout | M1 |
Reduce to the given form and justify the change in limits | A1 | [3]
**(ii)** Convert integrand to a sum of integrable terms and attempt integration | M1 |
Obtain integral $\frac{1}{2}\ln u + \frac{1}{u} - \frac{1}{4u^2}$, or equivalent | A1 + A1 |
(deduct A1 for each error or omission)
Substitute limits in an integral containing two terms of the form $a \ln u$ and $bu^{-2}$ | M1 |
Obtain answer $\frac{1}{2}\ln 2 - \frac{7}{16}$, exact simplified equivalent | A1 | [5]
---7 Let $I = \int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { \left( 1 + x ^ { 2 } \right) ^ { 3 } } \mathrm {~d} x$.\\
(i) Using the substitution $u = 1 + x ^ { 2 }$, show that $I = \int _ { 1 } ^ { 2 } \frac { ( u - 1 ) ^ { 2 } } { 2 u ^ { 3 } } \mathrm {~d} u$.\\
(ii) Hence find the exact value of $I$.
\hfill \mbox{\textit{CAIE P3 2016 Q7 [8]}}