Standard +0.3 This is a straightforward integration by parts question with definite integral evaluation. While it requires careful execution of a standard technique and exact arithmetic with special angles (π/6 and π/3), it involves no conceptual difficulty or problem-solving insight—just methodical application of a core C4/Year 2 technique. The 6 marks reflect the working required rather than conceptual challenge, making it slightly easier than average.
Given that
$$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} x \cos x \, dx = a\pi + b$$
find the exact value of \(a\) and the exact value of \(b\).
Fully justify your answer.
[6 marks]
Question 8:
8 | Uses integration by parts with
u= xand v′=cosx
xsinx+cosx
PI by | 3.1a | B1 | u = x u′=1
v′=cosx v=sinx
∫ xcosxdx=xsinx−∫
sinxdx
=xsinx+cosx
π
π
∫3 xcosxdx=[xsinx+cosx]3
π π
4 4
π π π π π π
= sin +cos − sin +cos
3 3 3 4 4 4
3 1 2 2
=π + −π +
6 2 8 2
4 3−3 2 1− 2
= π+
24 2
Applies integration by parts
formula correctly by substituting
their u, u’, v and v’
xsinx+cosx
PI by | 1.1a | M1
xsinx+cosx
Obtains
CAO | 1.1b | A1
Substitutes limits correctly into
their integrated expression
PI by correct a and b | 1.1a | M1
Uses correct exact value for any
π 3 π 1
one of sin = or cos =
3 2 3 2
or
π 2 π 2
cos = or sin =
4 2 4 2
PI by correct a or b | 1.2 | B1
Obtains correct exact values of
a and b
ACF
Ignore if 0.14(…) seen
subsequently | 1.1b | A1
Total | 6
Q | Marking instructions | AO | Marks | Typical solution
Given that
$$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} x \cos x \, dx = a\pi + b$$
find the exact value of $a$ and the exact value of $b$.
Fully justify your answer.
[6 marks]
\hfill \mbox{\textit{AQA Paper 3 2021 Q8 [6]}}