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QuestionArt
Find the indefinite integral of f(2$) = (3x^2 - 2x + 5) / (x^3 + 4x). Show all steps of partial fraction decomposition and explain your approach. Include a final answer at the end that clearly states the complete solution.
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Answer
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Step 1: Understand the integral and the function
We want to find the indefinite integral of the function First, note that the denominator can be factored to simplify the expression before integrating.
Step 2: Factor the denominator
Factor out the common factor 1$ cannot be factored further over the real numbers since it has no real roots.
Final Answer
\boxed{ \int \frac{3x^2 - 2x + 5}{x^3 + 4x} dx = \frac{5}{4} \ln|x| + \frac{7}{8} \ln(x^2 + 4) - \arctan \left( \frac{x}{2} \right) + C. }
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