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GUJCET · Maths · Integrals
\(\int_0^1 \frac{d x}{(3 x+2)+\sqrt{3 x+2}}=\) _________.
- A \(-\frac{2}{3} \log \left|\frac{\sqrt{5}+1}{\sqrt{2}+1}\right|\)
- B \(2 \log |\sqrt{5}+1|\)
- C \(\frac{2}{3} \log \left|\frac{\sqrt{5}+1}{\sqrt{2}+1}\right|\)
- D \(\frac{2}{3} \log |\sqrt{5}+1|\)
Answer & Solution
Correct Answer
(C) \(\frac{2}{3} \log \left|\frac{\sqrt{5}+1}{\sqrt{2}+1}\right|\)
Step-by-step Solution
Detailed explanation
Let \( u = \sqrt{3x+2} \). Then \( u^2 = 3x+2 \implies 2u \, du = 3 \, dx \implies dx = \frac{2}{3}u \, du \). Limits: \( x=0 \implies u=\sqrt{2} \); \( x=1 \implies u=\sqrt{5} \).
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