Find $\int x \cos x \, dx$.

(C) To evaluate $\int x \cos x \, dx$,we use the integration by parts formula: $\int f(x)g(x) \, dx = f(x) \int g(x) \, dx - \int [f'(x) \int g(x) \, dx] \, dx$.
Using the $LIATE$ rule,we choose $f(x) = x$ (algebraic) and $g(x) = \cos x$ (trigonometric).
Applying the formula:
$\int x \cos x \, dx = x \int \cos x \, dx - \int [\frac{d}{dx}(x) \int \cos x \, dx] \, dx$
$= x \sin x - \int (1) \sin x \, dx$
$= x \sin x - (-\cos x) + C$
$= x \sin x + \cos x + C$
If we had chosen $f(x) = \cos x$ and $g(x) = x$,the integral would become more complex,demonstrating the importance of the $LIATE$ rule.