Testing for convergence or divergence of an integral: Use the direct comparison test or limit comparison test to test the integral for convergence.
integral (0 to 1) of dt/(t-sint)
How do I know which test to use, and how should I proceed from there?

Question

Testing for convergence or divergence of an integral: Use the direct comparison test or limit comparison test to test the integral for convergence.
integral (0 to 1) of dt/(t-sint)
How do I know which test to use, and how should I proceed from there?

rogue · Answer

I wish I had my calculator on me :( Try comparing it to 1/t.

anonymous · Answer

does it matter whether I compare it to 1/t versus 1/t^2 or 1/t^3, etc?

rogue · Answer

Yep, this diverges ;)

1/(t - sin t) > 1/t from t:[0,1]
$$\int\limits_{0}^{1} \frac {dt}{t} = \left[ \ln t \right]_{0}^{1}$$ That is divergent. Since 1/t (the smaller function) diverges, then 1/(t- sin t) (the bigger function) must diverge as well.

rogue · Answer

It doesn't matter for this case, but for other integrals, you might wanna choose carefully.

anonymous · Answer

I see. thanks. Could I use LCT for this integral 
as well?