CUET · MATHS · PYQ PAPER 2025
Value of \(\int \frac{2}{(x-3) \sqrt{x+1}} d x\) is : (Here \(C\) is an arbitrary constant)
- A \(\log \left|\frac{x-3}{x+1}\right|+C\)
- B \(\log \left|\frac{\sqrt{x-1}-2}{\sqrt{x+1}}\right|+C\)
- C \(\frac{1}{2} \log \left|\frac{\sqrt{x+1}-1}{\sqrt{x+1}+1}\right|+C\)
- D \(\log \left|\frac{\sqrt{x+1-2}}{\sqrt{x+1}+2}\right|+C\)
Answer & Solution
Correct Answer
(D) \(\log \left|\frac{\sqrt{x+1-2}}{\sqrt{x+1}+2}\right|+C\)
Step-by-step Solution
Detailed explanation
Let \(u = \sqrt{x+1} \implies u^2 = x+1 \implies x = u^2-1 \implies dx = 2u \, du\). \(\int \frac{2}{(u^2-1-3)u} (2u \, du) = \int \frac{4}{u^2-4} du\) \(= \int \left( \frac{1}{u-2} - \frac{1}{u+2} \right) du\) \(= \ln|u-2| - \ln|u+2| + C\)…
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