QQuestionMathematics
QuestionMathematics
Find the indefinite integral of f(2$) = (3x^2 - 2x + 5) / (x^3 + 4x). Show all steps of partial fraction decomposition and explain your approach. Include a final answer at the end that clearly states the complete solution.
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Answer
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Step 1:: Understand the integral and the function
We want to find the indefinite integral of the function First, note that the denominator can be factored to simplify the expression before integrating.
Step 2:: Factor the denominator
Factor out the common factor 1$ cannot be factored further over the real numbers since it has no real roots.
Step 3:: Set up partial fraction decomposition
Since the denominator factors as 1$ are constants to be determined.
Step 4:: Multiply both sides by the denominator to clear fractions
Multiply both sides by 1$: 3x^2 - 2x + 5 = A(x^2 + 4) + (Bx + C)(x).
Step 5:: Expand the right side
Expand each term: A(x^2 + 4) = A x^2 + 4A, and (Bx + C)(x) = B x^2 + C x. So, 3x^2 - 2x + 5 = A x^2 + 4A + B x^2 + C x.
Step 6:: Group like terms on the right side
Group terms by powers of 1$: 3x^2 - 2x + 5 = (A + B) x^2 + C x + 4A.
Step 7:: Equate coefficients of corresponding powers of 1$
Match the coefficients of 1$.
Step 8:: Solve the system of equations
From the constant term: From the coefficient of 1$:
Step 9:: Write the partial fraction decomposition with the found constants
Step 10:: Write the integral in terms of partial fractions
The integral becomes:
Step 11:: Integrate each term separately
First integral: Second integral:
Step 12:: Evaluate the integral 1$
Use substitution: Therefore,
Step 13:: Evaluate the integral 1$
Recall the standard integral: Here, 1$, so
Step 14:: Substitute back into the integral expression
Putting it all together:
Step 15:: Write the complete integral solution
Combine all parts: where 1$ is the constant of integration. ---
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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