\( \int \frac { x ^ { 2 } + 20 } { x ^ { 5 } - 2 x ^ { 2 } + 5 x } d x \)

It seems that the integral I attempted to calculate is not supported directly. Let's break down the integral step by step and see if we can simplify it or find an alternative method to solve it. The integral we want to evaluate is: \[ \int \frac{x^2 + 20}{x^5 - 2x^2 + 5x} \, dx \] ### Step 1: Factor the Denominator First, let's factor the denominator \(x^5 - 2x^2 + 5x\). We can factor out \(x\): \[ x^5 - 2x^2 + 5x = x(x^4 - 2x + 5) \] Now, we need to analyze the polynomial \(x^4 - 2x + 5\) to see if it can be factored further. ### Step 2: Analyze the Polynomial The polynomial \(x^4 - 2x + 5\) does not have real roots (as can be checked using the discriminant or by evaluating it at various points). Therefore, we will keep it as is. ### Step 3: Partial Fraction Decomposition Next, we can use partial fraction decomposition to rewrite the integrand: \[ \frac{x^2 + 20}{x(x^4 - 2x + 5)} = \frac{A}{x} + \frac{Bx^3 + Cx^2 + Dx + E}{x^4 - 2x + 5} \] ### Step 4: Set Up the Equation Multiplying both sides by the denominator \(x(x^4 - 2x + 5)\) gives: \[ x^2 + 20 = A(x^4 - 2x + 5) + (Bx^3 + Cx^2 + Dx + E)x \] ### Step 5: Expand and Collect Like Terms Now we will expand the right-hand side and collect like terms. This will allow us to equate coefficients for \(x^4\), \(x^3\), \(x^2\), \(x\), and the constant term. Let's proceed with this expansion and then solve for \(A\), \(B\), \(C\), \(D\), and \(E\). ### Step 6: Solve for Coefficients After expanding, we will have a system of equations based on the coefficients of \(x\). Let's calculate the coefficients. 1. The coefficient of \(x^4\) gives us \(A + B = 0\). 2. The coefficient of \(x^3\) gives us \(D = 0\). 3. The coefficient of \(x^2\) gives us \(C = A + 1\). 4. The coefficient of \(x\) gives us \(-2A + E = 0\). 5. The constant term gives us \(5A = 20\). Now, let's solve these equations. ### Step 7: Solve the System of Equations We can solve the equations: 1. \(5A = 20\) implies \(A = 4\). 2. From \(A + B = 0\), we have \(B = -4\). 3. From \(C = A + 1\), we have \(C = 5\). 4. From \(D = 0\), we have \(D = 0\). 5. From \(-2A + E = 0\), we have \(E = 8\). Thus, we have: - \(A = 4\) - \(B = -4\) - \(C = 5\) - \(D = 0\) - \(E = 8\) ### Step 8: Rewrite the Integral Now we can rewrite the integral as: \[ \int \left( \frac{4}{x} + \frac{-4x^3 + 5x^2 + 8}{x^4 - 2x + 5} \right) \, dx \] ### Step 9: Integrate Each Term Now we can integrate each term separately: 1. \(\int \frac{4}{x} \, dx = 4 \ln |x|\) 2. For the second term, we will need to perform polynomial long division or further integration techniques. Let's calculate the integral of the second term. ### Step 10: Calculate the Integral of the Second Term We will need to evaluate: \[ \int \frac{-4x^3 + 5x^2 + 8}{x^4 - 2x + 5} \, dx \] This may require additional techniques such as substitution or numerical methods. ### Conclusion The integral can be expressed as: \[ 4 \ln |x| + \int \frac{-4x^3 + 5x^2 + 8}{x^4 - 2x + 5} \, dx + C \] Where \(C\) is the constant of integration. If you would like to proceed with the integration of the second term or need further assistance, please let me know!