Question:
\(\int \frac{x}{5-x^2} \, dx =\)
\(\int \frac{x}{5-x^2} \, dx =\)
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For \(\int \frac{x}{a - x^2} dx\), substitute \(u = a - x^2\).
Updated On: Apr 24, 2026
- \(-\frac{1}{2} \log_e \left|\frac{1}{5-x^2}\right| + C\)
- \(\frac{1}{2} \log_e |5-x^2| + C\)
- \(-2 \log_e |5-x^2| + C\)
- \(-\frac{1}{2} \log_e |5-x^2| + C\)
- \(2 \log_e \left|\frac{1}{5-x^2}\right| + C\)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
Use substitution \(u = 5 - x^2\). Then \(du = -2x \, dx \Rightarrow x \, dx = -\frac{du}{2}\).
Step 2: Detailed Explanation:
\(\int \frac{x}{5-x^2} dx = \int \frac{1}{u} \cdot \left(-\frac{du}{2}\right) = -\frac{1}{2} \int \frac{du}{u} = -\frac{1}{2} \ln|u| + C = -\frac{1}{2} \ln|5 - x^2| + C\)
Step 3: Final Answer:
\(-\frac{1}{2} \log_e |5 - x^2| + C\).
Use substitution \(u = 5 - x^2\). Then \(du = -2x \, dx \Rightarrow x \, dx = -\frac{du}{2}\).
Step 2: Detailed Explanation:
\(\int \frac{x}{5-x^2} dx = \int \frac{1}{u} \cdot \left(-\frac{du}{2}\right) = -\frac{1}{2} \int \frac{du}{u} = -\frac{1}{2} \ln|u| + C = -\frac{1}{2} \ln|5 - x^2| + C\)
Step 3: Final Answer:
\(-\frac{1}{2} \log_e |5 - x^2| + C\).
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