Split the integrand into partial fractions.
![\dfrac{9x+2}{x^2+x-6} = \dfrac{9x+2}{(x-2)(x+3)} = \dfrac a{x-2} + \dfrac b{x+3}](https://tex.z-dn.net/?f=%5Cdfrac%7B9x%2B2%7D%7Bx%5E2%2Bx-6%7D%20%3D%20%5Cdfrac%7B9x%2B2%7D%7B%28x-2%29%28x%2B3%29%7D%20%3D%20%5Cdfrac%20a%7Bx-2%7D%20%2B%20%5Cdfrac%20b%7Bx%2B3%7D)
![\implies 9x+2 = a(x+3) + b(x-2) = (a+b)x + (3a-2b)](https://tex.z-dn.net/?f=%5Cimplies%209x%2B2%20%3D%20a%28x%2B3%29%20%2B%20b%28x-2%29%20%3D%20%28a%2Bb%29x%20%2B%20%283a-2b%29)
![\implies \begin{cases}a+b=9 \\ 3a-2b=2\end{cases} \implies a=4,b=5](https://tex.z-dn.net/?f=%5Cimplies%20%5Cbegin%7Bcases%7Da%2Bb%3D9%20%5C%5C%203a-2b%3D2%5Cend%7Bcases%7D%20%5Cimplies%20a%3D4%2Cb%3D5)
Then we have
![\displaystyle \int \frac{9x+2}{x^2+x-6} \, dx = 4 \int \frac{dx}{x-2} + 5 \int \frac{dx}{x+3} \\\\ = \boxed{4\ln|x-2| + 5\ln|x+3| + C}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%5Cfrac%7B9x%2B2%7D%7Bx%5E2%2Bx-6%7D%20%5C%2C%20dx%20%3D%204%20%5Cint%20%5Cfrac%7Bdx%7D%7Bx-2%7D%20%2B%205%20%5Cint%20%5Cfrac%7Bdx%7D%7Bx%2B3%7D%20%5C%5C%5C%5C%20%3D%20%5Cboxed%7B4%5Cln%7Cx-2%7C%20%2B%205%5Cln%7Cx%2B3%7C%20%2B%20C%7D)
which follows from the result
![\displaystyle \int \frac{dx}x = \ln|x|+C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%5Cfrac%7Bdx%7Dx%20%3D%20%5Cln%7Cx%7C%2BC)
Answer: c I think
Step-by-step explanation:
Answer:
(-42/67,-231/67)
Step-by-step explanation:
-21=-21=-21
Answer:
8x-18x+27x=17x
Step-by-step explanation:
Answer:
10/5 - 4
Step-by-step explanation:
you have to write down what you understand from the sentence.
First, you read it carefully since it says quotient then you will have to put that first and then subtraction. Normally you would put subtraction in front because the sentence is in that order. But since your dealing with division then the subtraction part would go afer.
10/5 - 4