Question

Consider the improper integrals
$I_1 = \int_{1}^{\infty} \frac{t \sin t}{e^t} \, dt$
and
$I_2 = \int_{1}^{\infty} \frac{1}{\sqrt{t}} \ln\left(1 + \frac{1}{t}\right) \, dt.$
Then:

• A. $I_1$ converges, but $I_2$ does NOT converge.
• B. $I_1$ does NOT converge, but $I_2$ converges.
• C. Both $I_1$ and $I_2$ converge.
• D. Neither $I_1$ nor $I_2$ converges.

Answer 1

Answer: (C)

Both $I_1$ and $I_2$ converge.

Answer 2

The correct option is (C): Both $I_1$ and $I_2$ converge