Integration of $\int_{0}^{D/2} \frac{2R^2 x}{(R^2 + x^2)^2} \, dx$ is:
Options:
(A) R²/(R² + D²)
(B) 4R²/(4R² + D²)
(C) D²/(R² + D²)
(D) D²/(4R² + D²)
Official Answer Key:
D
Step-by-Step Solution & Explanation
Put u = R² + x² ⇒ du = 2x dx. Limits: x = 0 ⇒ u = R²; x = D/2 ⇒ u = R² + D²/4.
I = R² ∫ du/u² = R²[−1/u] = R²(1/R² − 1/(R² + D²/4)).
= 1 − 4R²/(4R² + D²) = (4R² + D² − 4R²)/(4R² + D²) = D²/(4R² + D²).
Distractor (B) is the term that must be subtracted from 1, not the answer itself.
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