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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 integral ∫ 5 x x - 2 dx = x + a | x - 2 x| + k , then the...

If the integral ∫ \dfrac{5 \text\tan \text{x}}{\text\tan \text{x} - 2} \text{dx} = \text{x} + \text{a} \text\log \left|\text\sin \text{x} - 2 \text\cos \text{x}\right| + \text{k}, then the value of aa is

Options

  1. A11
  2. B22
  3. C−1- 1 & & & & &
  4. D−2- 2 & & & & &

Correct answer

B. 22

Step-by-step solution

∫ \dfrac{5 \text\tan \text{x}}{\text\tan \text{x} - 2} \text{dx} = \text{x} + \text{a} \text\ln \left|\text\sin \text{x} - 2 \text\cos \text{x}\right| + \text{k}

Differentiating both sides, we get

\dfrac{5 \text\tan \text{x}}{\text\tan \text{x} - 2} = 1 + \dfrac{\text{a} \left(\text\cos \text{x} + 2 \text\sin \text{x}\right)}{\text\sin \text{x} - 2 \text\cos \text{x}}

⇒ \dfrac{5 \text\sin \text{x}}{\text\sin \text{x} - 2 \text\cos \text{x}} = \dfrac{\text\sin \text{x} \left(1 + 2 \text{a}\right) + \text\cos \text{x} \left(\text{a} - 2\right)}{\text\sin \text{x} - 2 \text\cos \text{x}}

⇒a=2⇒ \text{a} = 2
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