QQuestionMathematics
QuestionMathematics
Find the indefinite integral of f(2$) = (3x^2 - 2x + 5) / (x^3 + 4x). Show all steps of partial fraction decomposition and explain your approach. Include a final answer at the end that clearly states the complete solution.
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Step 1:To find the indefinite integral of the function
3x^2 - 2x + 5 = (A + B)x^2 + Cx + 4A.
we will use partial fraction decomposition. Let's break this down step by step. ### Step 1: Factor the Denominator First, we need to factor the denominator 1$: Expanding the right-hand side gives: Combining like terms results in: ### Step 4: Set Up a System of Equations Now we can equate the coefficients from both sides:
Step 2:
For 1$
Step 3:
For 1$
Step 4:
\frac{3x^2 - 2x + 5}{x(x^2 + 4)} = \frac{5/4}{x} + \frac{(7/4)x - 2}{x^2 + 4}.
For the constant term: 1$ back into our partial fractions: ### Step 7: Integrate Each Term Now we can integrate each term separately:
Step 5:
\int \frac{5/4}{x} \, dx = \frac{5}{4} \ln |x| + C_1.
For the first term:
Step 6:
\int f(x) \, dx = \frac{5}{4} \ln |x| + \frac{7}{8} \ln |x^2 + 4| - \frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right) + C,
For the second term, we can split it into two parts: For the first part, we use the substitution 1$ is the constant of integration. ###
Final Answer
The indefinite integral of 1$ is: \int f(x) \, dx = \frac{5}{4} \ln |x| + \frac{7}{8} \ln |x^2 + 4| - \frac{1}{2} \tan^{- 1}\left(\frac{x}{2}\right) + C.
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