CUET · MATHS · PYQ PAPER 2025
For \(x>e\),\(\int \frac{d x}{x-\sqrt{x}}\) is equal to
- A \(-2 \log _e|\sqrt{x}-1|+C : C\) is an arbitrary constant
- B \(4 \log _e|\sqrt{x}-1|+C : C\) is an arbitrary constant
- C \(\log _e|\sqrt{x}-1|+C : C\) is an arbitrary constant
- D \(2 \log _e|\sqrt{x}-1|+C : C\) is an arbitrary constant
Answer & Solution
Correct Answer
(D) \(2 \log _e|\sqrt{x}-1|+C : C\) is an arbitrary constant
Step-by-step Solution
Detailed explanation
Let \(u=\sqrt{x}\). Then \(u^2=x\) and \(dx=2u\,du\). \(\int \frac{2u\,du}{u^2-u} = \int \frac{2u\,du}{u(u-1)}\) \(= \int \frac{2\,du}{u-1} = 2 \log_e|u-1|+C\) \(= 2 \log_e|\sqrt{x}-1|+C\)
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