| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Show definite integral equals specific value (requiring partial fractions or complex algebra) |
| Difficulty | Standard +0.3 This is a straightforward two-part integration question requiring a standard substitution u=√x (with clear guidance), followed by algebraic manipulation to split the integrand and routine integration. The substitution mechanics are explicit, and the final integration uses polynomial division/splitting which is a standard C3/P3 technique. Slightly above average only due to the algebraic manipulation needed in part (ii), but well within typical exam expectations. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(du = \frac{1}{2\sqrt{x}}dx\) | B1 | |
| Substitute for \(x\) and \(dx\) throughout | M1 | |
| Justify the change in limits and obtain the given answer | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Convert integrand into the form \(A + \frac{B}{u+1}\) | M1* | |
| Obtain integrand \(A=1\), \(B=-2\) | A1 | |
| Integrate and obtain \(u - 2\ln(u+1)\) | A1\(\checkmark\) + A1\(\checkmark\) | |
| Substitute limits correctly in integral containing terms \(au\) and \(b\ln(u+1)\), where \(ab \neq 0\) | DM1 | |
| Obtain the given answer following full and correct working | A1 | [6] The f.t. is on \(A\) and \(B\) |
## Question 6:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $du = \frac{1}{2\sqrt{x}}dx$ | B1 | |
| Substitute for $x$ and $dx$ throughout | M1 | |
| Justify the change in limits and obtain the given answer | A1 | [3] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Convert integrand into the form $A + \frac{B}{u+1}$ | M1* | |
| Obtain integrand $A=1$, $B=-2$ | A1 | |
| Integrate and obtain $u - 2\ln(u+1)$ | A1$\checkmark$ + A1$\checkmark$ | |
| Substitute limits correctly in integral containing terms $au$ and $b\ln(u+1)$, where $ab \neq 0$ | DM1 | |
| Obtain the given answer following full and correct working | A1 | [6] The f.t. is on $A$ and $B$ |
---6 Let $I = \int _ { 1 } ^ { 4 } \frac { ( \sqrt { } x ) - 1 } { 2 ( x + \sqrt { } x ) } \mathrm { d } x$.\\
(i) Using the substitution $u = \sqrt { } x$, show that $I = \int _ { 1 } ^ { 2 } \frac { u - 1 } { u + 1 } \mathrm {~d} u$.\\
(ii) Hence show that $I = 1 + \ln \frac { 4 } { 9 }$.
\hfill \mbox{\textit{CAIE P3 2016 Q6 [9]}}