JEE Mains · Maths · STD 12 - 7.2 definite integral
The value of the integral \(\displaystyle\int_{0}^{2} \dfrac{\sqrt{x(x^2+x+1)}}{(\sqrt{x+1})(\sqrt{x^4+x^2+1})} \, dx\) is equal to:
- A \(\dfrac{1}{3}\log_e(3-2\sqrt{2})\)
- B \(\dfrac{2}{3}\log_e(4+\sqrt{2})\)
- C \(\dfrac{2}{3}\log_e(3+2\sqrt{2})\)
- D \(\dfrac{1}{3}\log_e(1+6\sqrt{2})\)
Answer & Solution
Correct Answer
(C) \(\dfrac{2}{3}\log_e(3+2\sqrt{2})\)
Step-by-step Solution
Detailed explanation
Let \(I = \displaystyle\int_{0}^{2} \dfrac{\sqrt{x(x^2 + x + 1)}}{\sqrt{x + 1}\,\sqrt{x^4 + x^2 + 1}}\, dx\). Using the identity \(x^4 + x^2 + 1 = (x^2 + x + 1)(x^2 - x + 1)\):…
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