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 above, 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.

Answer 2

To find the values of \(a\) and \(b\), we need to compare the given integral expression with the expression 

\[ a \log |1 + x^2| + b \tan^{-1}(x) + \frac{1}{5} \log |x + 2| + c. \]

By 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 the coefficients of the corresponding terms must match. Therefore, we have:

\[ a = -\frac{1}{10} \quad \text{and} \quad b = \frac{2}{5}. \]

Thus, the correct answer is option (A)

\[ \boxed{a = -\frac{1}{10} \quad \text{and} \quad b = \frac{2}{5}}. \]