| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Topic | Integration by Substitution |
| Type | Square root substitution: definite integral |
| Difficulty | Standard +0.8 Part (a) is straightforward recall of exponential differentiation. Part (b) requires recognizing a substitution (u = 2^x or u = 3 + 2^x), correctly handling the chain rule in reverse, changing limits, and evaluating a power integral—multiple coordinated steps with some technical care needed for the exact answer. This is moderately challenging but within reach of a well-prepared FM student using standard integration techniques. |
| Spec | 1.06b Gradient of e^(kx): derivative and exponential model1.08h Integration by substitution |
\begin{enumerate}[label=(\alph*)]
\item Given that $u = 2^x$, write down an expression for $\frac{du}{dx}$
[1 mark]
\item Find the exact value of $\int_0^1 2^x \sqrt{3 + 2^x} dx$
Fully justify your answer.
[6 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q11 [7]}}