| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| 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 clear guidance. Part (i) requires routine manipulation of the given substitution to transform the integral, while part (ii) involves expanding and integrating a polynomial divided by u, then evaluating limits. The substitution is provided, and both parts are 'show that' questions with target answers given, making it slightly easier than average but requiring careful algebraic manipulation. |
| Spec | 1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(du = -\frac{1}{2\sqrt{x}}dx\), or equivalent | B1 | |
| Substitute for \(x\) and \(dx\) throughout | M1 | |
| Obtain integrand \(\frac{\pm 2(2-u)^2}{u}\), or equivalent | A1 | |
| Show correct working to justify the change in limits and obtain the given answer with no errors seen | A1 | [4] |
| (ii) Integrate and obtain at least two terms of the form \(a\ln u\), \(bu\), and \(cu^2\) | M1✱ | |
| Obtain indefinite integral \(8\ln u - 8u + u^2\), or equivalent | A1 | |
| Substitute limits correctly | M1(dep✱) | |
| Obtain the given answer correctly having shown sufficient working | A1 | [4] |
**(i)** State or imply $du = -\frac{1}{2\sqrt{x}}dx$, or equivalent | B1 |
Substitute for $x$ and $dx$ throughout | M1 |
Obtain integrand $\frac{\pm 2(2-u)^2}{u}$, or equivalent | A1 |
Show correct working to justify the change in limits and obtain the given answer with no errors seen | A1 | [4]
**(ii)** Integrate and obtain at least two terms of the form $a\ln u$, $bu$, and $cu^2$ | M1✱ |
Obtain indefinite integral $8\ln u - 8u + u^2$, or equivalent | A1 |
Substitute limits correctly | M1(dep✱) |
Obtain the given answer correctly having shown sufficient working | A1 | [4]6 Let $I = \int _ { 0 } ^ { 1 } \frac { \sqrt { } x } { 2 - \sqrt { } x } \mathrm {~d} x$.\\
(i) Using the substitution $u = 2 - \sqrt { } x$, show that $I = \int _ { 1 } ^ { 2 } \frac { 2 ( 2 - u ) ^ { 2 } } { u } \mathrm {~d} u$.\\
(ii) Hence show that $I = 8 \ln 2 - 5$.
\hfill \mbox{\textit{CAIE P3 2015 Q6 [8]}}