Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Split the integrand into partial fractions.
[tex]\dfrac{9x+2}{x^2+x-6} = \dfrac{9x+2}{(x-2)(x+3)} = \dfrac a{x-2} + \dfrac b{x+3}[/tex]
[tex]\implies 9x+2 = a(x+3) + b(x-2) = (a+b)x + (3a-2b)[/tex]
[tex]\implies \begin{cases}a+b=9 \\ 3a-2b=2\end{cases} \implies a=4,b=5[/tex]
Then we have
[tex]\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}[/tex]
which follows from the result
[tex]\displaystyle \int \frac{dx}x = \ln|x|+C[/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
