Question

If \(\int \frac{dx}{(x+2)(x^2 + 1)} = a \log |1+x^2| + b \tan^{-1} x + \frac{1}{5} \log |x+2| + c\), then

• A. \(a = -\frac{1}{10}\) ,\( b = \frac{2}{5}\)
• B. \(a = \frac{-1}{10} , b = - \frac{2}{5}\)
• C. \(a =\frac{ 1}{10} , b = \frac{2}{5}\)
• D. \(a = \frac{1}{10} , b = -\frac{ 2}{5}\)

Answer 1

The Correct Option is A

To find the values of a and b, we can compare the given integral expression with the expression
\(a \log |1 + x^2| + b \tan^{-1}(x) + \frac{1}{5} \log |x + 2| + c\) 
Comparing the integrand of the given integral with the expression 
\(a \log |1 + x^2| + b \tan^{-1}(x) + \frac{1}{5} \log |x + 2| + c\), 
we can see that: 
\(a = -\frac{1}{10} \quad \text{and} \quad b = \frac{2}{5}\)
Therefore, option (A) \(a = -\frac{1}{10} \quad \text{and} \quad b = \frac{2}{5}\) is the correct answer.