Moderate -0.3 This is a straightforward integration by parts question with standard trigonometric functions and simple limits. While it requires knowing the integration by parts technique and careful execution, it's a routine C4 exercise with no conceptual challenges—slightly easier than average since it's a direct application of a core technique with clean limits.
\(x \sin x - \int \sin x \, dx = (x \sin x + \cos x)\)
M1, A1, B1, M1, A1 5
For attempt at parts going correct way (\(u = x\), \(dv = \cos x\) and \(f(x) +/- \int g(x) \, dx\)); For both terms correct; Indic anywhere that \(\int \sin x \, dx = -\cos x\); For correct method of limits; For correct exact answer ISW
Answer = \(\frac{1}{2}\pi - 1\)
$x \sin x - \int \sin x \, dx = (x \sin x + \cos x)$ | M1, A1, B1, M1, A1 5 | For attempt at parts going correct way ($u = x$, $dv = \cos x$ and $f(x) +/- \int g(x) \, dx$); For both terms correct; Indic anywhere that $\int \sin x \, dx = -\cos x$; For correct method of limits; For correct exact answer **ISW**
Answer = $\frac{1}{2}\pi - 1$ | |