| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Independent multi-part (different techniques) |
| Difficulty | Standard +0.3 This is a standard C4 question testing integration by parts (routine technique with given answer to verify), volume of revolution (direct formula application), and product rule differentiation. All parts use well-practiced techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07q Product and quotient rules: differentiation1.08i Integration by parts4.08d Volumes of revolution: about x and y axes |
\begin{enumerate}[label=(\alph*)]
\item Use integration by parts to show that
$$\int_0^{\frac{\pi}{4}} x \sec^2 x \, dx = \frac{1}{4}\pi - \frac{1}{2} \ln 2.$$ [6]
\end{enumerate}
\includegraphics{figure_1}
The finite region $R$, bounded by the equation $y = x^{\frac{1}{2}} \sec x$, the line $x = \frac{\pi}{4}$ and the $x$-axis is shown in Fig. 1. The region $R$ is rotated through $2\pi$ radians about the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the volume of the solid of revolution generated. [2]
\item Find the gradient of the curve with equation $y = x^{\frac{1}{2}} \sec x$ at the point where $x = \frac{\pi}{4}$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q6 [11]}}