Discussion
\(\int \frac{x}{5 - x^{2}}dx =\)
Answer: D
The integral is \(-\frac{1}{2}\log_{e}|5 - x^{2}| + C\).
The derivative of \(5 - x^{2}\) is \(-2x\), so write the numerator as \(-\frac{1}{2}(-2x)\):
\[ \begin{aligned} \int\frac{x}{5 - x^{2}}dx \\ &= -\frac{1}{2}\int\frac{-2x}{5 - x^{2}}dx \\ &= -\frac{1}{2}\log|5 - x^{2}| + C \end{aligned} \]
The option \(\frac{1}{2}\log|5 - x^{2}|\) has the wrong sign; note that \(-\frac{1}{2}\log\left|\frac{1}{5 - x^{2}}\right|\) equals that same wrong expression. The options with coefficient \(\pm 2\) do not differentiate back to the integrand.
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