# Definite Integral

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

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1. 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 button-pushing.

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2. alright, thanks Reiny!

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3. 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)

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4. Thanks Steve!

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