A

Step 1: Evaluate the first integral
We have:
$$ I_1 = \int (\sin x + \frac{1}{x} + 2 \frac{1}{x^2} + 3x^3) \, dx $$
Integrate term by term:
- For \(\sin x\): \(\int \sin x \, dx = -\cos x + C_1\)
- For \(\frac{1}{x}\): \(\int \frac{1}{x} \, dx = \ln|x| + C_2\)
- For \(2 \frac{1}{x^2}\): \(\int 2 \frac{1}{x^2} \, dx = -\frac{2}{x} + C_3\)
- For \(3x^3\): \(\int 3x^3 \, dx = \frac{3}{4}x^4 + C_4\)
Combining all terms:
$$ I_1 = -\cos x + \ln|x| - \frac{2}{x} + \frac{3}{4}x^4 + C $$

Step 2: Evaluate the second integral
We have:
$$ I_2 = \int (3\cos x + e^x + 4x^2 + x + 5) \, dx $$
Integrate term by term:
- For \(3\cos x\): \(\int 3\cos x \, dx = 3\sin x + C_5\)
- For \(e^x\): \(\int e^x \, dx = e^x + C_6\)
- For \(4x^2\): \(\int 4x^2 \, dx = \frac{4}{3}x^3 + C_7\)
- For \(x\): \(\int x \, dx = \frac{1}{2}x^2 + C_8\)
- For \(5\): \(\int 5 \, dx = 5x + C_9\)
Combining all terms:
$$ I_2 = 3\sin x + e^x + \frac{4}{3}x^3 + \frac{1}{2}x^2 + 5x + C $$

After evaluation, we obtain the desired results.
Therefore, option A is selected as it contains the correct integration results.