Which of the following inequalities is/are TRUE?

First evaluate each definite integral exactly by using integration by parts, then compare its numerical value (in radians) with the number given in the option.

Case A: $$I_1=\int_{0}^{1} x\cos x\,dx$$

Take $$u=x,\;dv=\cos x\,dx\;$$ so $$du=dx,\;v=\sin x.$$
$$I_1=x\sin x-\int\sin x\,dx=x\sin x+\cos x+C.$$
Therefore,
$$I_1=\bigl[x\sin x+\cos x\bigr]_{0}^{1}=(\sin 1+\cos 1)-1=\sin 1+\cos 1-1.$$
Numerically, $$\sin 1\approx0.841471,\;\cos 1\approx0.540302,$$ hence
$$I_1\approx0.841471+0.540302-1=0.381773.$$
Since $$\dfrac38=0.375,$$ we have $$I_1\gt\dfrac38,$$ so Option A is TRUE.

Case B: $$I_2=\int_{0}^{1} x\sin x\,dx$$

Take $$u=x,\;dv=\sin x\,dx\;$$ so $$du=dx,\;v=-\cos x.$$
$$I_2=-x\cos x+\int\cos x\,dx=-x\cos x+\sin x+C.$$
Thus,
$$I_2=\bigl[-x\cos x+\sin x\bigr]_{0}^{1}=\sin 1-\cos 1.$$
Using the same numerical values,
$$I_2\approx0.841471-0.540302=0.301169.$$
Because $$\dfrac{3}{10}=0.300000,$$ we get $$I_2\gt\dfrac{3}{10},$$ hence Option B is TRUE.

Case C: $$I_3=\int_{0}^{1} x^{2}\cos x\,dx$$

First integration by parts: $$u=x^{2},\;dv=\cos x\,dx$$ gives $$du=2x\,dx,\;v=\sin x.$$
$$I_3=x^{2}\sin x-\int 2x\sin x\,dx.$$
For $$\int x\sin x\,dx$$ we already have the formula $$-x\cos x+\sin x.$$ Hence
$$I_3=x^{2}\sin x-2(-x\cos x+\sin x)=x^{2}\sin x+2x\cos x-2\sin x.$$
Now
$$I_3=\bigl[x^{2}\sin x+2x\cos x-2\sin x\bigr]_{0}^{1}=2\cos 1-\sin 1.$$
Numerically,
$$I_3\approx2(0.540302)-0.841471=1.080604-0.841471=0.239133.$$
Since Option C claims $$I_3\ge1,$$ which is false, Option C is NOT true.

Case D: $$I_4=\int_{0}^{1} x^{2}\sin x\,dx$$

Again, let $$u=x^{2},\;dv=\sin x\,dx$$ so $$du=2x\,dx,\;v=-\cos x.$$
$$I_4=-x^{2}\cos x+\int 2x\cos x\,dx.$$
Using $$\int x\cos x\,dx=x\sin x+\cos x,$$ we get
$$I_4=-x^{2}\cos x+2x\sin x+2\cos x.$$
Hence
$$I_4=\bigl[-x^{2}\cos x+2x\sin x+2\cos x\bigr]_{0}^{1}=(\cos 1+2\sin 1)-2.$$
Numerically,
$$I_4\approx0.540302+2(0.841471)-2=0.540302+1.682942-2=0.223244.$$
Since Option D claims $$I_4\ge\dfrac38=0.375,$$ which is false, Option D is NOT true.

Therefore the correct statements are:
Option A ($$\int_{0}^{1} x\cos x\,dx\ge\dfrac38$$) and Option B ($$\int_{0}^{1} x\sin x\,dx\ge\dfrac{3}{10}$$).

Final Answer: Option A and Option B.