Questions: Express the integrand as a sum of partial fractions and evaluate the integral. [ int fracd x4-9 x^2 ]

Express the integrand as a sum of partial fractions and evaluate the integral. [ int fracd x4-9 x^2 ]
Transcript text: Express the integrand as a sum of partial fractions and evaluate the integral. \[ \int \frac{d x}{4-9 x^{2}} \]
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Solution

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Solution Steps

Step 1: Factor the Denominator

We start with the integral \[ \int \frac{d x}{4 - 9 x^{2}}. \] The denominator can be factored as \[ 4 - 9 x^{2} = (2 - 3x)(2 + 3x). \]

Step 2: Partial Fraction Decomposition

Next, we express the integrand as a sum of partial fractions: \[ \frac{1}{4 - 9 x^{2}} = \frac{1}{4(3x + 2)} - \frac{1}{4(3x - 2)}. \]

Step 3: Integrate Each Term

We then integrate each term separately: \[ \int \left( \frac{1}{4(3x + 2)} - \frac{1}{4(3x - 2)} \right) d x. \] This results in: \[ -\frac{1}{12} \log |3x - 2| + \frac{1}{12} \log |3x + 2| + C, \] where \(C\) is the constant of integration.

Final Answer

\(\boxed{-\frac{1}{12} \log |3x - 2| + \frac{1}{12} \log |3x + 2| + C}\)

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