Trig Sub Homework: Did I Make a Mistake?

[Feodalherren]

Homework Statement

Did I make a mistake here somewhere? The solution in the back of the book is completely different. Seems like they used trig sub one step later or something. I can't find any error in my logic. Test coming up soon and I'm confused and panicking -_-!
Actually I just found a mistake with my constants! But apart from that, is it correct? The first term should be multiplied with -1/2 and the second with 3/2.

Homework Equations

The Attempt at a Solution

 

[Dick]

I doubt it's right, but it's too hard to read to make sure. I'd really suggest you write (x+4)=(x+1)+3 at the beginning and use that to split the integral into two much more manageable parts.

 

[Feodalherren]

I don't see how that makes it more manageable. That was my original plan but the numerator isn't my problem. No matter what I do I'm stuck with this denominator that's ugly.

 

[Dick]

I don't see how that makes it more manageable. That was my original plan but the numerator isn't my problem. No matter what I do I'm stuck with this denominator that's ugly.

The numerator is your problem. (x+1)/((x+1)^2+4) is an easy substitution. 3/((x+1)^2+4) is an easy arctan problem after the correct substitution. Divide and conquer. Don't try to do it all in one lump.

 

[ehild]

simplify the cos(arctan((x+1)/2)) term.

ehild

 

[Dick]

simplify the cos(arctan((x+1)/2)) term.

ehild

My point was not that you can't salvage that attempt by correcting a few errors, but that there is an easier strategy to do it to begin with.

 

[ehild]

My point was not that you can't salvage that attempt by correcting a few errors, but that there is an easier strategy to do it to begin with.

For me, the solution looks good (when correcting the constants) but the OP can bring it to a simpler form. He is familiar with the trigonometric substitution, and he will do it anyway when integrating 1/((x+1)²+4). You discourage the OP suggesting to discard his method.

ehild

 

[Dick]

For me, the solution looks good (when correcting the constants) but the OP can bring it to a simpler form. He is familiar with the trigonometric substitution, and he will do it anyway when integrating 1/((x+1)²+4). You discourage the OP suggesting to discard his method.

ehild

I do discourage the OP from doing it that way. Not because it can't be made correct with some corrections to constants (and they do need to be corrected). But because if you split it up cleverly to begin with you don't have to simplify cos(arctan) type stuff and you make a lot fewer mistakes. You can power though with a single substitution, but why? Why are you fighting my suggestion??