| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Find curve from gradient |
| Difficulty | Moderate -0.3 This is a straightforward C1 integration question requiring students to integrate a simple expression (polynomial plus power of x), find the constant of integration using a given condition, then evaluate at a specific point. While it involves multiple steps and algebraic manipulation, all techniques are routine for this level with no conceptual challenges or novel problem-solving required. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f(x) = \int (5 + \frac{4}{x^2}) \, dx\) | ||
| \(f(x) = 5x - 4x^{-1} + c\) | M1 A2 | |
| (b) \(f(1) = 5 - 4 + c = 1 + c\) | M1 | |
| \(f(2) = 10 - 2 + c = 8 + c\) | ||
| \(f(2) = 2f(1)\) \(\therefore 8 + c = 2(1 + c)\) | M1 | |
| \(c = 6\) | A1 | |
| \(f(x) = 5x - 4x^{-1} + 6\) | ||
| \(f(4) = 20 - 1 + 6 = 25\) | M1 A1 | (8) |
**(a)** $f(x) = \int (5 + \frac{4}{x^2}) \, dx$ | |
$f(x) = 5x - 4x^{-1} + c$ | M1 A2 |
**(b)** $f(1) = 5 - 4 + c = 1 + c$ | M1 |
$f(2) = 10 - 2 + c = 8 + c$ | |
$f(2) = 2f(1)$ $\therefore 8 + c = 2(1 + c)$ | M1 |
$c = 6$ | A1 |
$f(x) = 5x - 4x^{-1} + 6$ | |
$f(4) = 20 - 1 + 6 = 25$ | M1 A1 | (8)Given that
$$\text{f}'(x) = 5 + \frac{4}{x^2}, \quad x \neq 0,$$
\begin{enumerate}[label=(\alph*)]
\item find an expression for $\text{f}(x)$. [3]
\end{enumerate}
Given also that
$$\text{f}(2) = 2\text{f}(1),$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find $\text{f}(4)$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q7 [8]}}