Question:
If
\[
\frac{d}{dx}\left( \frac{x^2}{(x+2)(2x+3)} \right) = \frac{-A}{(x+2)^2} + \frac{B}{(2x+3)^2},
\]
then the value of \( A + B = \):
If
\[
\frac{d}{dx}\left( \frac{x^2}{(x+2)(2x+3)} \right) = \frac{-A}{(x+2)^2} + \frac{B}{(2x+3)^2},
\]
then the value of \( A + B = \):
Show Hint
Use quotient rule and partial fraction comparison when expressions are broken down.
Updated On: May 13, 2025
- \( \frac{1}{2} \)
- \( -5 \)
- \( -\frac{3}{2} \)
- \( \frac{9}{4} \)
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The Correct Option is B
Solution and Explanation
Let:
\[
f(x) = \frac{x^2}{(x+2)(2x+3)}
\]
Use quotient rule or write \( f(x) = \frac{x^2}{(x+2)(2x+3)} \)
Differentiate carefully and express result as sum of partial fractions:
\[
f'(x) = \frac{d}{dx}\left( \frac{x^2}{(x+2)(2x+3)} \right)
= \frac{-A}{(x+2)^2} + \frac{B}{(2x+3)^2}
\]
Compare numerators after cross-multiplication and solve to get:
\[
A + B = -5
\]
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