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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Indefinite Integration

NTA Abhyas 2020 · Mathematics · Indefinite Integration

If the value of integral ∫ dx (x + x^ 2 - 1 ) ^ 2 = ax^ 3 - x + b...

If the value of integral ∫dx(x+x2−1)2=ax3−x+b(x2−1)1b+C∫ \dfrac{dx}{{\left(x + \sqrt{x^{2} - 1}\right)}^{2}} = ax^{3} - x + b{\left(x^{2} - 1\right)}^{\dfrac{1}{b}} + C, (where, CC is the constant of integration), then a×ba \times b& is equal to

Options

  1. A11
  2. B49\dfrac{4}{9}
  3. C22
  4. D94\dfrac{9}{4}

Correct answer

B. 49\dfrac{4}{9}

Step-by-step solution

∫(x−x2−1)2dx=∫x2+(x2−1)−2xx2−1dx∫ {\left(x - \sqrt{x^{2} - 1}\right)}^{2}dx = ∫ x^{2} + \left(x^{2} - 1\right) - 2x\sqrt{x^{2} - 1}dx
=2x33−x+23(x2−1)32+C⇒a=b=23= \dfrac{2x^{3}}{3} - x + \dfrac{2}{3}{\left(x^{2} - 1\right)}^{\dfrac{3}{2}} + C⇒ a = b = \dfrac{2}{3}
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