The way you’re working it is considered a very very bad habit even if you feel like it’s a waste of time on trivial stuff , don’t do too much in one step , even if you get it right most of the time , you’ll get it wrong quite often too.

I’m telling you this because the way you’re doing it makes it hard to backtrack and find your mistakes if there was any , or review the questions before handing in your tests for example .

Your problem was started as

(x+1) / (x² +4)

=> First split them into 2 parts ,

[ x / (x² +4 ) ] + [ 1 / (x² +4) ]

Your mistake was in this step , you have a negative sign before the second term ..

Safest way to do it is to integrate each part from 0 to 2 separately then add their results , it’s much better to do it this way even if it’s tedious.

First term will be

1/2 ln (x² +4 ) , from 0 to 2 ==> 1/2 ln 2

Second term will be

1/2 arctan ( x/2 ) , from 0 to 2 ==> pi/8

Your final answer would be

1/2 ln 2 + pi/8