Integration math problem

can someone me with this question?

0 to ∞ ∫(arctan(x))/(2+e^x)dx Evaluate and determine if its convergent or divergent using comparison theorem?

 

That integral would be a headache to evaluate directly - even using a computer, it looks like a mess. Just looking at the graph of the function though, I suppose you could estimate it with a riemann sum from 0 to about 5.

To show it's convergent, though:
arctan(x) < Pi/2 for all x

So 0 to &#8734; &#8747;(arctan(x))/(2+e^x)dx < 0 to &#8734; &#8747;(Pi/2)/(2+e^x)dx

We know 0 to &#8734; &#8747;(Pi/2)/(2+e^x)dx = -Pi/4 * e^(-x) evaluated from 0 to &#8734;
-Pi/4 * e^(-x) evaluated from 0 to &#8734; = 0 - (-Pi/4) = Pi/4
So 0 to &#8734; &#8747;(Pi/2)/(2+e^x)dx = Pi/4

Thus, &#8747;(arctan(x))/(2+e^x)dx < Pi/4, so it's convergent.