Question

If \[ \int \frac{2x^2+3}{(x^2-1)(x^2-4)} \, dx = \log \left[ \left( \frac{x-2}{x+2} \right)^a \cdot \left( \frac{x+1}{x-1} \right)^b \right] + c \] where \( c \) is the constant of integration, then the value of \[ a+b = \, ? \] 

• A. \(\frac{1}{12}\)
• B. \(\frac{21}{12}\)
• C. \(\frac{-1}{12}\)
• D. \(\frac{-21}{12}\)

Hint:

If the final answer is in logarithmic form with factors like \[ \left(\frac{x-a}{x+a}\right)^m, \] then partial fractions is almost always the right method.

Answer 1

The Correct Option is B

Concept:
We use partial fractions and then compare the resulting logarithmic expression with the given form. ip

Step 1: Resolve into partial fractions.
\[ \frac{2x^2+3}{(x^2-1)(x^2-4)} = \frac{A}{x-2}+\frac{B}{x+2}+\frac{C}{x-1}+\frac{D}{x+1} \] Solving, we get: \[ A=\frac{11}{12},\quad B=-\frac{11}{12},\quad C=-\frac{5}{6},\quad D=\frac{5}{6} \] So, \[ \frac{2x^2+3}{(x^2-1)(x^2-4)} = \frac{11}{12(x-2)}-\frac{11}{12(x+2)}-\frac{5}{6(x-1)}+\frac{5}{6(x+1)} \] ip

Step 2: Integrate term by term.
\[ \int \frac{2x^2+3}{(x^2-1)(x^2-4)}dx = \frac{11}{12}\log|x-2|-\frac{11}{12}\log|x+2| -\frac{5}{6}\log|x-1|+\frac{5}{6}\log|x+1|+c \] ip

Step 3: Write in the required form.
\[ = \log\left[\left(\frac{x-2}{x+2}\right)^{11/12} \left(\frac{x+1}{x-1}\right)^{5/6}\right]+c \] So, \[ a=\frac{11}{12},\qquad b=\frac{5}{6}=\frac{10}{12} \] Therefore, \[ a+b=\frac{11}{12}+\frac{10}{12}=\frac{21}{12} \] ip Hence, the correct answer is:
\[ \boxed{(B)\ \frac{21}{12}} \]