| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Trigonometric substitution: show transformation then evaluate |
| Difficulty | Standard +0.3 This is a standard C4 integration question using a given substitution. Part (i) requires routine application of the substitution x = tan θ with dx = sec²θ dθ and algebraic simplification using trig identities. Part (ii) involves integrating cos²θ using the double angle formula and careful handling of limits. While it requires multiple steps and some care with the trigonometry, it follows a well-practiced template for substitution questions and uses standard C4 techniques throughout. Slightly easier than average due to the substitution being provided rather than requiring students to identify it themselves. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(dx = \sec^2\theta \, d\theta\) | AEF | M1 |
| Indefinite integral = \(\int \cos^2\theta \, d\theta\) | M1, A1 3 | |
| (ii) \(k \left[ \frac{\theta}{2} + \frac{1}{4}\cos 2\theta \right]\) | M1, A1 4 | |
| Limits = \(\frac{1}{4}\pi(\text{accept 45}) \) and \(0\) | ||
| \(\frac{(\pi + 2)}{8}\) | AEF | |
| Single 'parts' + \(\sin^2\theta = 1 - \cos^2\theta\) acceptable |
(i) $dx = \sec^2\theta \, d\theta$ | **AEF** | M1 | Attempt to connect $dx,d\theta$ (not $dx = d\theta$)
Indefinite integral = $\int \cos^2\theta \, d\theta$ | | M1, A1 3 | For $dx = \sec^2\theta \, d\theta$ or equiv correctly used; With at least one intermed step **AG**
(ii) $k \left[ \frac{\theta}{2} + \frac{1}{4}\cos 2\theta \right]$ | | M1, A1 4 | "Satis" attempt to change to double angle; For correct attempt + correct integration; New limits for $\theta$ or resubstituting; Ignore decimals after correct answer
Limits = $\frac{1}{4}\pi(\text{accept 45}) $ and $0$ | |
$\frac{(\pi + 2)}{8}$ | | **AEF**
Single 'parts' + $\sin^2\theta = 1 - \cos^2\theta$ acceptable | |\begin{enumerate}[label=(\roman*)]
\item Show that the substitution $x = \tan \theta$ transforms $\int \frac{1}{(1 + x^2)^2} dx$ to $\int \cos^2 \theta d\theta$. [3]
\item Hence find the exact value of $\int_0^1 \frac{1}{(1 + x^2)^2} dx$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C4 2005 Q4 [7]}}