int(2x^(3)+7x^(2)+7x+9)/((x^(2)+x+1)(x^(2)+x+2))dx

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Decomposition Method: To solve the integral of the given rational function, we will use the method of partial fraction decomposition. This method involves expressing the integrand as a sum of simpler fractions whose denominators are the factors of the original denominator.Decompose Integrands: First, we need to decompose the integrand into partial fractions. We assume that the integrand can be written in the form: (2x^(3)+7x^(2)+7x+9)/((x^(2)+x+1)(x^(2)+x+2)) = A/(x^(2)+x+1) + B/(x^(2)+x+2) where A and B are constants to be determined.Find Constants A and B: To find the constants A and B, we multiply both sides of the equation by the common denominator (x^(2)+x+1)(x^(2)+x+2) to get: 2x^(3)+7x^(2)+7x+9 = A(x^(2)+x+2) + B(x^(2)+x+1)Solve for A and B: Now we need to solve for A and B. To do this, we can either compare coefficients or plug in convenient values for x to create a system of equations. Let's choose to plug in values for x that will simplify the equation. We can choose x = -1 and x = -2, which will zero out one of the terms on the right side in each case.Plug in Values: Plugging in x = -1, we get: 2(-1)^(3)+7(-1)^(2)+7(-1)+9 = A((-1)^(2)+(-1)+2) -2+7-7+9 = A(1-1+2) 7 = 2A A = 7/2Calculate A and B: Plugging in x = -2, we get: 2(-2)^(3)+7(-2)^(2)+7(-2)+9 = B((-2)^(2)+(-2)+1) -16+28-14+9 = B(4-2+1) 7 = 3B B = 7/3Rewrite Integrands: Now that we have the values for A and B, we can rewrite the integrand as: (2x^(3)+7x^(2)+7x+9)/((x^(2)+x+1)(x^(2)+x+2)) = (7/2)/(x^(2)+x+1) + (7/3)/(x^(2)+x+2)Integrate Separately: We can now integrate each term separately. The integrals do not correspond to elementary functions, so we will need to use a substitution or another method to integrate them. However, we notice that the numerators are not the derivatives of the denominators, which is usually a requirement for a straightforward integration of rational functions. This suggests that we may have made a mistake in our partial fraction decomposition.

$\int \frac{2 x^{3}+7 x^{2}+7 x+9}{\left(x^{2}+x+1\right)\left(x^{2}+x+2\right)} d x$

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∫2x3+7x2+7x+9(x2+x+1)(x2+x+2)dx \int \frac{2 x^{3}+7 x^{2}+7 x+9}{\left(x^{2}+x+1\right)\left(x^{2}+x+2\right)} d x ∫(x2+x+1)(x2+x+2)2x3+7x2+7x+9​dx

$\int \frac{2 x^{3}+7 x^{2}+7 x+9}{\left(x^{2}+x+1\right)\left(x^{2}+x+2\right)} d x$