| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2018 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Partial fractions after substitution |
| Difficulty | Standard +0.3 This is a standard C4 integration question testing three routine techniques: partial fractions (with repeated linear factor), expanding and integrating a binomial, and substitution. Part (i) uses the cover-up method for partial fractions, part (ii) is straightforward binomial expansion, and part (iii) follows a given substitution hint. All are textbook exercises requiring competent technique but no novel insight, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08b Integrate x^n: where n != -1 and sums1.08h Integration by substitution1.08j Integration using partial fractions |
3. (i) Given that
$$\frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \equiv \frac { A } { ( 2 x + 1 ) } + \frac { B } { ( 2 x + 1 ) ^ { 2 } } + \frac { C } { ( x + 3 ) }$$
\begin{enumerate}[label=(\alph*)]
\item find the values of the constants $A , B$ and $C$.
\item Hence find
$$\int \frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \mathrm { d } x , \quad x > - \frac { 1 } { 2 }$$
(ii) Find
$$\int \left( \mathrm { e } ^ { x } + 1 \right) ^ { 3 } \mathrm {~d} x$$
(iii) Using the substitution $u ^ { 3 } = x$, or otherwise, find
$$\int \frac { 1 } { 4 x + 5 x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x , \quad x > 0$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2018 Q3 [14]}}