Discussion
Integration of \(\int_{0}^{D/2} \frac{2R^{2}x\,dx}{(R^{2} + x^{2})^{2}}\) is
Answer: D
The integral equals \(\frac{D^{2}}{4R^{2} + D^{2}}\).
Since \(\frac{d}{dx}(R^{2} + x^{2}) = 2x\), the antiderivative of \(\frac{2x}{(R^{2} + x^{2})^{2}}\) is \(-\frac{1}{R^{2} + x^{2}}\):
\[ \begin{aligned} I &= R^{2}\left[-\frac{1}{R^{2} + x^{2}}\right]_{0}^{D/2} \\ &= R^{2}\left(\frac{1}{R^{2}} - \frac{4}{4R^{2} + D^{2}}\right) \\ &= 1 - \frac{4R^{2}}{4R^{2} + D^{2}} = \frac{D^{2}}{4R^{2} + D^{2}} \end{aligned} \]
\(\frac{4R^{2}}{4R^{2} + D^{2}}\) is the subtracted term alone, and the options with \(R^{2} + D^{2}\) use \(D\) instead of \(D/2\) as the upper limit.
No comment is present. Be the first to comment.
Loading…
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development