Solve: int(x^(3)+1)/(x^(3)-5x^(2)+6x)dx.

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Simplify the integrand: Simplify the integrand if possible. The integrand is a rational function. We can attempt to simplify it by factoring the denominator and seeing if there is a common factor that can be canceled with the numerator. The denominator is x^3 - 5x^2 + 6x. We can factor out an x to get x(x^2 - 5x + 6). Now, we factor the quadratic: x^2 - 5x + 6 = (x - 2)(x - 3). So, the denominator becomes x(x - 2)(x - 3). The numerator is x^3 + 1, which does not have any common factors with the denominator. Therefore, the integrand cannot be simplified by canceling common factors.Perform polynomial long division: Perform polynomial long division if the degree of the numerator is greater than or equal to the degree of the denominator. Since the degree of the numerator (3) is equal to the degree of the denominator (3), we can perform polynomial long division to simplify the integrand. However, in this case, the numerator is x^3 + 1, and the denominator is x^3 - 5x^2 + 6x. The leading terms of the numerator and denominator are the same (x^3), so when we divide x^3 by x^3, we get 1. The remainder will be the original numerator minus the product of the quotient (1) and the denominator, which is x^3 + 1 - (x^3 - 5x^2 + 6x). This simplifies to 5x^2 - 6x + 1. So, the integral can be rewritten as the integral of 1 plus the integral of (5x^2 - 6x + 1) / (x^3 - 5x^2 + 6x).Break into partial fractions: Break the integral into partial fractions. We have the integral of 1 plus the integral of (5x^2 - 6x + 1) / (x(x - 2)(x - 3)). To integrate the second part, we need to express it as a sum of partial fractions. We assume that (5x^2 - 6x + 1) / (x(x - 2)(x - 3)) can be written as A/x + B/(x - 2) + C/(x - 3). Multiplying both sides by the denominator, we get 5x^2 - 6x + 1 = A(x - 2)(x - 3) + Bx(x - 3) + Cx(x - 2). We will need to solve for A, B, and C.Solve for A, B, and C: Solve for A, B, and C. To find A, B, and C, we can plug in values for x that simplify the equation. For A, let x = 0: 1 = A(-2)(-3) + 0 + 0, so A = 1/6. For B, let x = 2: 5(4) - 6(2) + 1 = 0 + 2B(0) + 2C(0), which does not give us a value for B or C. This is a mistake; we need to choose values for x that will eliminate two of the terms to solve for the remaining variable.

Solve: $\int\frac{x^{3}+1}{x^{3}-5x^{2}+6x}\,dx$ .

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Solve: $\int\frac{x^{3}+1}{x^{3}-5x^{2}+6x}\,dx$ .