Standard +0.8 This question requires students to identify that the integrand has a discontinuity at x=π/4 within the integration interval (where cos x = sin x makes the denominator zero), demonstrating understanding that improper integrals arise from interior discontinuities, not just endpoint issues. It tests conceptual understanding of improper integrals beyond routine computation, requiring critical analysis of a flawed argument.
A student states that \(\int_0^{\pi/2} \frac{\cos x + \sin x}{\cos x - \sin x} \, dx\) is not an improper integral because \(\frac{\cos x + \sin x}{\cos x - \sin x}\) is defined at both \(x = 0\) and \(x = \frac{\pi}{2}\)
Assess the validity of the student's argument.
[2 marks]
Question 4:
4 | Correctly states that the student’s
argument is invalid because the
integrand is undefined at a value in
π
the interval (0, )
2 | AO2.3 | E1 | π
When x = ,
4
cosx−sinx =0
∴the integrand is undefined at this
point and the integral is improper
Correctly justifies the reason why it
is undefined and states a correct
conclusion | AO2.4 | E1
Total | 2
Q | Marking Instructions | AO | Marks | Typical Solution
A student states that $\int_0^{\pi/2} \frac{\cos x + \sin x}{\cos x - \sin x} \, dx$ is not an improper integral because $\frac{\cos x + \sin x}{\cos x - \sin x}$ is defined at both $x = 0$ and $x = \frac{\pi}{2}$
Assess the validity of the student's argument.
[2 marks]
\hfill \mbox{\textit{AQA Further Paper 1 Q4 [2]}}