| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2014 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Indefinite integral with linear substitution |
| Difficulty | Moderate -0.8 Both parts are standard textbook exercises in integration by substitution/recognition. Part (a) requires recognizing a linear function raised to a power (immediate application of chain rule in reverse), and part (b) is a standard logarithmic form where the numerator is a constant multiple of the derivative of the denominator. These are routine C3/C4 questions requiring pattern recognition rather than problem-solving. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int(2x+3)^{12}\,dx = \frac{(2x+3)^{13}}{(13)(2)} + c\) | M1 | Gives \(\pm\lambda(2x+3)^{13}\) where \(\lambda\) is constant, or \(\pm\mu(x + \frac{3}{2})^{13}\) |
| \(\frac{(2x+3)^{13}}{(13)(2)} + c\) | A1 | Coefficient need not be simplified; ignore \(+c\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int\frac{5x}{4x^2+1}\,dt = \frac{5}{8}\ln(4x^2+1) + c\) or \(\frac{5}{8}\ln(x^2 + \frac{1}{4}) + k\) | M1 | Gives \(\pm\mu\ln(4x^2+1)\) where \(\mu\) is constant, or \(\pm\mu\ln(x^2+\frac{1}{4})\) or \(\pm\mu\ln(k(4x^2+1))\) |
| \(\frac{5}{8}\ln(4x^2+1) + c\) | A1 | Modulus sign not needed but allow \(\frac{5}{8}\ln\ |
# Question 4:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int(2x+3)^{12}\,dx = \frac{(2x+3)^{13}}{(13)(2)} + c$ | M1 | Gives $\pm\lambda(2x+3)^{13}$ where $\lambda$ is constant, or $\pm\mu(x + \frac{3}{2})^{13}$ |
| $\frac{(2x+3)^{13}}{(13)(2)} + c$ | A1 | Coefficient need not be simplified; ignore $+c$ |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int\frac{5x}{4x^2+1}\,dt = \frac{5}{8}\ln(4x^2+1) + c$ or $\frac{5}{8}\ln(x^2 + \frac{1}{4}) + k$ | M1 | Gives $\pm\mu\ln(4x^2+1)$ where $\mu$ is constant, or $\pm\mu\ln(x^2+\frac{1}{4})$ or $\pm\mu\ln(k(4x^2+1))$ |
| $\frac{5}{8}\ln(4x^2+1) + c$ | A1 | Modulus sign not needed but allow $\frac{5}{8}\ln\|4x^2+1\|$; also allow $0.625\ln(4x^2+1)$; condone lack of $+c$. Note: $\frac{5}{8}\ln 4x^2 + 1$ with no bracket scores M1A0 |
---4. Find
\begin{enumerate}[label=(\alph*)]
\item $\int ( 2 x + 3 ) ^ { 12 } \mathrm {~d} x$
\item $\int \frac { 5 x } { 4 x ^ { 2 } + 1 } \mathrm {~d} x$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 2014 Q4 [4]}}