Find the exact value of \(\int_1^2 {\left( {{{(x - 2)}^2} + \frac{1}{x} + \sin \pi x} \right){\text{dx}}} \).

\(\left[ {\frac{1}{3}{{(x - 2)}^3} + \ln x - \frac{1}{\pi }\cos \pi x} \right]_{(1)}^{(2)}\)     A1A1A1

Note: Accept \(\frac{1}{3}{x^3} - 2{x^2} + 4x\) in place of \(\frac{1}{3}{(x - 2)^3}\).

\( = \left( {0 + \ln 2 - \frac{1}{\pi }\cos 2\pi } \right) - \left( { - \frac{1}{3} + \ln 1 - \frac{1}{\pi }\cos \pi } \right)\)     (M1)

\( = \frac{1}{3} + \ln 2 - \frac{2}{\pi }\)     A1A1

Note: Award A1 for any two terms correct, A1 for the third correct.

[6 marks]

Generally well done, although quite a number of candidates were either unable to integrate the sine term or incorrectly evaluated the resulting cosine at the limits.