int(9x+2)/(x^(2)+x-6)dx

Factor Denominator: Factor the denominator of the integrand. The denominator x^2 + x - 6 can be factored into (x + 3)(x - 2). So, the integral becomes int (9x + 2) / ((x + 3)(x - 2)) dx.Partial Fraction Decomposition: Perform partial fraction decomposition. We want to express (9x + 2) / ((x + 3)(x - 2)) as A / (x + 3) + B / (x - 2). To find A and B, we set up the equation 9x + 2 = A(x - 2) + B(x + 3).Solve for A and B: Solve for A and B. To find A and B, we can use the method of equating coefficients or plugging in convenient values for x. Let's use the latter method. For x = 2, we get 9(2) + 2 = B(2 + 3), which gives us B = (18 + 2) / 5 = 4. For x = -3, we get 9(-3) + 2 = A(-3 - 2), which gives us A = (-27 + 2) / -5 = 5.Rewrite with Partial Fractions: Rewrite the integral using the partial fractions. Now that we have A = 5 and B = 4, the integral becomes int 5 / (x + 3) dx + int 4 / (x - 2) dx.Integrate Each Term: Integrate each term separately. The integral of 5 / (x + 3) dx is 5 * ln|x + 3|, and the integral of 4 / (x - 2) dx is 4 * ln|x - 2|.Combine and Add Constant: Combine the results and add the constant of integration. The final answer is 5 * ln|x + 3| + 4 * ln|x - 2| + C, where C is the constant of integration.

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$\int\frac{9x+2}{x^{2}+x-6}\,dx$

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Calculus

Evaluate definite integrals using the chain rule

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\int\frac{9x+2}{x^{2}+x-6}\,dx

Factor Denominator: Factor the denominator of the integrand. $\newline$ The denominator $x^2 + x - 6$ can be factored into $(x + 3)(x - 2)$ . $\newline$ So, the integral becomes $\int \frac{9x + 2}{(x + 3)(x - 2)} \, dx$ .
Partial Fraction Decomposition: Perform partial fraction decomposition. $\newline$ We want to express $(9x + 2) / ((x + 3)(x - 2))$ as $A / (x + 3) + B / (x - 2)$ . $\newline$ To find $A$ and $B$ , we set up the equation $9x + 2 = A(x - 2) + B(x + 3)$ .
Solve for A and B: Solve for A and B. $\newline$ To find $A$ and $B$ , we can use the method of equating coefficients or plugging in convenient values for $x$ . Let's use the latter method. $\newline$ For $x = 2$ , we get $9(2) + 2 = B(2 + 3)$ , which gives us $B = (18 + 2) / 5 = 4$ . $\newline$ For $x = -3$ , we get $9(-3) + 2 = A(-3 - 2)$ , which gives us $A = (-27 + 2) / -5 = 5$ .
Rewrite with Partial Fractions: Rewrite the integral using the partial fractions. $\newline$ Now that we have $A = 5$ and $B = 4$ , the integral becomes $\int \frac{5}{(x + 3)} \, dx + \int \frac{4}{(x - 2)} \, dx$ .
Integrate Each Term: Integrate each term separately. $\newline$ The integral of $\frac{5}{x + 3} \, dx$ is $5 \cdot \ln|x + 3|$ , and the integral of $\frac{4}{x - 2} \, dx$ is $$4$ \cdot \ln|x - $2$|.
Combine and Add Constant: Combine the results and add the constant of integration. The final answer is $5 \cdot \ln|x + 3| + 4 \cdot \ln|x - 2| + C$, where $C$ is the constant of integration.