| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Independent multi-part (different techniques) |
| Difficulty | Moderate -0.3 Part (a) is a straightforward logarithmic integration requiring simple substitution or recognition. Part (b) is a standard textbook application of integration by parts with no complications. Both are routine C3-level exercises requiring direct application of learned techniques with no problem-solving insight needed, making this slightly easier than average. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{1}{2}\ln(3+2x) + c\) | M1 A1 | M1 for \(k\ln(3+2x)\); A1 for correct coefficient \(\frac{1}{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(u = x\), \(\frac{dv}{dx} = \sin\frac{x}{2}\); \(\frac{du}{dx} = 1\), \(v = -2\cos\frac{x}{2}\) | M1 | Correct parts assignment and differentiation/integration |
| \(= -2x\cos\frac{x}{2} + \int 2\cos\frac{x}{2}\,dx\) | A1 | Correct expression after applying parts |
| \(= -2x\cos\frac{x}{2} + 4\sin\frac{x}{2} + c\) | A1 A1 | A1 for each term correct |
# Question 5:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{1}{2}\ln(3+2x) + c$ | M1 A1 | M1 for $k\ln(3+2x)$; A1 for correct coefficient $\frac{1}{2}$ |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $u = x$, $\frac{dv}{dx} = \sin\frac{x}{2}$; $\frac{du}{dx} = 1$, $v = -2\cos\frac{x}{2}$ | M1 | Correct parts assignment and differentiation/integration |
| $= -2x\cos\frac{x}{2} + \int 2\cos\frac{x}{2}\,dx$ | A1 | Correct expression after applying parts |
| $= -2x\cos\frac{x}{2} + 4\sin\frac{x}{2} + c$ | A1 A1 | A1 for each term correct |5
\begin{enumerate}[label=(\alph*)]
\item Find $\int \frac { 1 } { 3 + 2 x } \mathrm {~d} x$.
\item By using integration by parts, find $\int x \sin \frac { x } { 2 } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2011 Q5 [6]}}