Uh I've posted a couple times before on improper integrals but this time I'm just going to ask all the questions and kind of scenarios that I have. If you can help me with these questions the other questions in my textbook will be easier because its pretty much the same from there.

1) x e^-x^2 dx between the interval of infinity and -infinity

For this question, I saw an example of it on pauls notes and well so I see that when dealing with interval between infinities means that you have to split the integral up, am i correct? Thats my question for this.

2) x+1/x^2+2x dx between infinity and 1

For this question how would I integrate this, I forgot how to integrate fractions well I know that say x/2x+3x I think would be integrated by first doing x/2x+x/3x am I right? But I'm not sure how how its done when it has two variables at the top and bottom.

3) S e^-5x ds between infinity and 0

For this question I've seen the answer for it and it tells me to use integration by parts. Why do I have to do integration by parts for this, isn't S just a variable aren't I able to keep the S out and just integrate the e^-5x and solve? How would I do this question or attempt it?

4) ln x/x dx between infinity and t

For this question I don't know how to approach it all.

5) x2/9+x^6 dx between infinity and -infinity

Lastly for this question since its between the infinities im guessing you have to split them up but again I'm not sure how since I don't know how to integrate fractions such as these.

Thanks in advance guys, I really need your help

Sorry if i'm being dumb but I need all the steps i can take for these questions :P