| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Integration with substitution and partial fractions |
| Difficulty | Standard +0.3 This is a structured multi-part question that guides students through standard C4 techniques: partial fractions with repeated linear factors (routine), verifying a given substitution (mechanical differentiation and algebra), and applying results to evaluate a definite integral. While it requires multiple steps, each part is straightforward with clear signposting, making it slightly easier than the average A-level question. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08h Integration by substitution |
8 (i) Given that $\frac { 2 t } { ( t + 1 ) ^ { 2 } }$ can be expressed in the form $\frac { A } { t + 1 } + \frac { B } { ( t + 1 ) ^ { 2 } }$, find the values of the constants $A$ and $B$.\\
(ii) Show that the substitution $t = \sqrt { 2 x - 1 }$ transforms $\int \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x$ to $\int \frac { 2 t } { ( t + 1 ) ^ { 2 } } \mathrm {~d} t$.\\
(iii) Hence find the exact value of $\int _ { 1 } ^ { 5 } \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x$.
\hfill \mbox{\textit{OCR C4 2008 Q8 [11]}}