Posted by Paul on .
Calculate the following integral:
upper limit: 1
lower limit: 2
∫ (5x + 3) / (x^2 + 4x + 7) dx
Do I do a polynomial division for this one?
Thank you

Definite Integral 
Reiny,
Nasty integral
long division will get you nowhere.
It would have been a good idea if the fraction had been flipped
If the denominator would have factored we could have used "integration by partial fractions" , but no such luck ( x^2 + 4x + 7 has no real roots, so we can't form real factors)
I tried integration by parts and bailed out after a messy start, and went to good ol' reliable
WOLFRAM
http://integrals.wolfram.com/index.jsp?expr=%285x%2B3%29%2F+%28x%5E2+%2B+4x+%2B+7%29&random=false
(when they say log, they actually imply Ln )
now sub in the upper and lower values and good luck on the buttonpushing. 
Definite Integral 
Paul,
alright, thanks Reiny!

Definite Integral 
Steve,
The way to get their answer is to complete the square and then use trig substitution.
If you go to wolframalpha.com and type in
∫ (5x + 3) / (x^2 + 4x + 7) dx
then click the "Show Steps" button you can see how they did it. If you want the final numeric answer, type in
∫[2,1] (5x + 3) / (x^2 + 4x + 7) dx
to get log32  7pi/sqrt(27) 
Definite Integral 
Paul,
Thanks Steve!