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sashaice [31]
2 years ago
9

Calculus problem:

Mathematics
2 answers:
Oliga [24]2 years ago
7 0

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}

\implies 9x+2 = a(x+3) + b(x-2) = (a+b)x + (3a-2b)

\implies \begin{cases}a+b=9 \\ 3a-2b=2\end{cases} \implies a=4,b=5

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}

which follows from the result

\displaystyle \int \frac{dx}x = \ln|x|+C

Luden [163]2 years ago
4 0
Don’t think it’s possible
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