| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Multi-part questions combining substitution with curve/area analysis |
| Difficulty | Standard +0.3 This is a structured multi-part question that guides students through standard C3 techniques: product rule differentiation, second derivative test, substitution in integration, and integration by parts. While it requires multiple methods and careful algebraic manipulation, each step follows predictable patterns with clear signposting. The substitution and integration by parts are routine applications rather than requiring novel insight, making this slightly easier than average. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07q Product and quotient rules: differentiation1.08h Integration by substitution1.08i Integration by parts |
7 A curve is defined by the equation $y = 2 x \ln ( 1 + x )$.\\
(i) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence verify that the origin is a stationary point of the curve.\\
(ii) Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$, and use this to verify that the origin is a minimum point.\\
(iii) Using the substitution $u = 1 + x$, show that $\int \frac { x ^ { 2 } } { 1 + x } \mathrm {~d} x = \int \left( u - 2 + \frac { 1 } { u } \right) \mathrm { d } u$.
Hence evaluate $\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { 1 + x } \mathrm {~d} x$, giving your answer in an exact form.\\
(iv) Using integration by parts and your answer to part (iii), evaluate $\int _ { 0 } ^ { 1 } 2 x \ln ( 1 + x ) \mathrm { d } x$.
\hfill \mbox{\textit{OCR MEI C3 2008 Q7 [19]}}