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
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.
alright, thanks Reiny!
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)
Thanks Steve!
Yes, to solve this integral, you will need to perform a polynomial division in order to simplify the integrand. Here's how you can do it:
1. First, factorize the denominator:
x^2 + 4x + 7 = (x + 2)^2 + 3
2. Perform polynomial division by dividing the numerator (5x + 3) by the denominator (x^2 + 4x + 7). This will give you a quotient and a remainder.
(5x + 3) / (x^2 + 4x + 7) = Q(x) + R(x) / (x^2 + 4x + 7)
The quotient (Q(x)) is obtained by dividing the highest degree term of the numerator by the highest degree term of the denominator, which gives Q(x) = 0. Therefore, the quotient is 0.
3. The remainder (R(x)) is obtained by subtracting the product of the divisor and quotient from the numerator. Therefore:
R(x) = (5x + 3) - 0(x^2 + 4x + 7)
R(x) = 5x + 3
4. Rewrite the original integral using the quotient and remainder:
∫ (5x + 3) / (x^2 + 4x + 7) dx = ∫ Q(x) dx + ∫ R(x) / (x^2 + 4x + 7) dx
Since the quotient is 0, the integral simplifies to:
∫ (5x + 3) / (x^2 + 4x + 7) dx = ∫ (5x + 3) / (x^2 + 4x + 7) dx
Now you can proceed to find the integral of R(x) / (x^2 + 4x + 7). This can be done using techniques like completing the square or partial fractions, depending on the complexity of the integrand.