Question

Evaluate the indefinite integral of the following partial fraction decomposition:

-2x^2 + 2x + 12/(x + 2)(x^2 + 4)

a) (-1/2ln|x + 2| - left(x/2) + C)

b) (1/2ln|x + 2| + left(x/2) + C)

c) (-ln|x + 2| - 2(x/2) + C)

d) (ln|x + 2| + 2(x/2) + C)

Answer 1

The correct indefinite integral of the given partial fraction decomposition
`\(-2x^2 + 2x + 12/(x + 2)(x^2 + 4)\) is \(-(1)/(2)\ln|x + 2| - (x)/(2) + C\),` where
`\(C\)` is the constant of integration (Option a).

Step-by-step explanation:

To evaluate the indefinite integral of the given partial fraction decomposition, we first decompose the fraction into partial fractions. The decomposition involves expressing the expression
`\(-2x^2 + 2x + 12\)` as the sum of three fractions with undetermined coefficients over the factors
`\(x + 2\) and \(x^2 + 4\).` Solving for the coefficients and integrating each term separately, we arrive at the result
`\(-(1)/(2)\ln|x + 2| - (x)/(2) + C\),` where
`\(C\)`represents the constant of integration.

The integration process involves standard techniques for integrating rational functions, including partial fraction decomposition. After obtaining the decomposition and determining the coefficients, each term is integrated individually. The natural logarithmic function arises from the integral of
`\((1)/(x + 2)\),` and the integration of
`\((1)/(x^2 + 4)\)` leads to an arctangent function. The resulting expression is then simplified to the correct form
`\(-(1)/(2)\ln|x + 2| - (x)/(2) + C\).`

In summary, (Option a).the evaluation of the indefinite integral involves the decomposition of the given rational function into partial fractions, determination of coefficients, and subsequent integration of each term. The final result is expressed in the specified form, providing the correct solution to the problem.