Standard +0.3 This is a straightforward substitution question where the substitution is explicitly given. Students must find dt/dx, rearrange to express dx in terms of dt, substitute into the integral, and integrate e^(2t). While it requires careful algebraic manipulation and understanding of the substitution method, it follows a standard template with no conceptual surprises, making it slightly easier than average.
\(\sqrt{x+1}\,e^{2\sqrt{x+1}}-\frac{1}{2}e^{2\sqrt{x+1}}+c\) cao www
A1
\(+c\) may be seen in previous line only for A1; if \(dt\) not seen in integral at some point impose penalty of 1 mark from total mark of 2 or more
# Question 5:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dt}{dx}=k(x+1)^{-1/2}$ or $\frac{dx}{dt}=2t$ from $x=t^2\pm1$ oe | M1 | Or e.g. $k\,dt=\frac{dx}{\sqrt{x+1}}$ oe |
| $\int kte^{2t}\,dt$ | M1* | $k$ is any non-zero constant |
| $kt\times\frac{1}{2}e^{2t}\pm k\int\frac{1}{2}e^{2t}\,dt$ | M1dep* | |
| $te^{2t}-\int e^{2t}\,dt$ | A1 | May be implied by the next A1 |
| $te^{2t}-\frac{1}{2}e^{2t}$ | A1 | |
| $\sqrt{x+1}\,e^{2\sqrt{x+1}}-\frac{1}{2}e^{2\sqrt{x+1}}+c$ cao www | A1 | $+c$ may be seen in previous line only for **A1**; if $dt$ not seen in integral at some point impose penalty of 1 mark from total mark of 2 or more |
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