I am an Exernal student (no in-person tuition or lectures), and I am practicing problems from my Calculus Study Guide, but the answers are not given. I'd appreciate if you could check my answers.

1. Determine whether the following integrals converge:

1.1

Type 2 improper integral - problem spot: x=0

Since , compare the original integral to which we know converges near 0 as the power of the denominator is less than 1.

Since , which converges, the original integral also converges.

1.2

Type 2 improper integral - problem spot: x=0.

Choose a function which is similar to the original integrand since

converges, and so does the original integral.

1.3

Type 1 improper integral - problem spot: infinity.

Chose a comparision function

as the denominator goes to infinity

Since converges, so does the original integral, by one-way limit test.

1.4

Type 2 improper integral - problem spot: x=0.

Here I am a bit lost and would appreciate some hints. I know that the denominator is bounded (ln (x+1) ranges from 0 to ln2, and x ranges from 0 to 1. Does it help to chose a convergence test?

+I will post more in a follow up post.