Which of the following inequalities is/are TRUE?

$\int_{0}^{1} x\cos x\,dx,\quad \int_{0}^{1} x\sin x\,dx$ $\int_{0}^{1} x^{2}\cos x\,dx,\quad \int_{0}^{1} x^{2}\sin x\,dx$ $\text{To find: which of (1)--(4) are TRUE}$ $\text{Plan: bound each integrand on } [0,1] \text{ using standard inequalities}$ $\cos x \geq 1 - \dfrac{x^{2}}{2},\qquad \sin x \geq x - \dfrac{x^{3}}{6}$ $\dfrac{d}{dx}\!\left(\cos x - 1 + \tfrac{x^{2}}{2}\right) = x - \sin x \geq 0 \Rightarrow \cos x \geq 1 - \tfrac{x^{2}}{2}$ $\dfrac{d}{dx}\!\left(\sin x - x + \tfrac{x^{3}}{6}\right) = \cos x - 1 + \tfrac{x^{2}}{2} \geq 0 \Rightarrow \sin x \geq x - \tfrac{x^{3}}{6}$ $x\cos x \geq x - \dfrac{x^{3}}{2} \Rightarrow \int_{0}^{1} x\cos x\,dx \geq \left[\dfrac{x^{2}}{2} - \dfrac{x^{4}}{8}\right]_{0}^{1} = \dfrac{1}{2} - \dfrac{1}{8} = \dfrac{3}{8}$ $\Rightarrow (1) \text{ is TRUE}$ $x\sin x \geq x^{2} - \dfrac{x^{4}}{6} \Rightarrow \int_{0}^{1} x\sin x\,dx \geq \dfrac{1}{3} - \dfrac{1}{30} = \dfrac{3}{10}$ $\Rightarrow (2) \text{ is TRUE}$ $x^{2}\cos x \leq x^{2} \Rightarrow \int_{0}^{1} x^{2}\cos x\,dx \leq \dfrac{1}{3} < \dfrac{1}{2}$ $\Rightarrow (3) \text{ is FALSE}$ $x^{2}\sin x \geq x^{3} - \dfrac{x^{5}}{6} \Rightarrow \int_{0}^{1} x^{2}\sin x\,dx \geq \dfrac{1}{4} - \dfrac{1}{36} = \dfrac{2}{9}$ $\Rightarrow (4) \text{ is TRUE}$ $\boxed{\text{TRUE: (1), (2), (4)}}$