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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 ∫ (ln x)/(x3)dx = f(x) 4x^ 2 + C , where f (e) = -...

If the integral ∫lnxx3dx=f(x)4x2+C∫ \dfrac{ln x}{x3}dx =\dfrac{f\left(x\right)}{4x^{2}} + C, where f(e)=−3f \left(e\right) = - 3 and& CC is the constant of integration,& then the value of& f(e2)f\left(e^{2}\right)& is equal to

Options

  1. A33
  2. B−4- 4
  3. C−5- 5
  4. D55

Correct answer

C. −5- 5

Step-by-step solution

∫lnx⏟I1x3⏟IIdx=(lnx)(−12x2)−∫1x(−12x2)dx∫ \mathop{\mathop{ln x}_{⏟}}_{I}\mathop{\mathop{\dfrac{1}{x^{3}}}_{⏟}}_{II}dx = \left(ln x\right)\left(\dfrac{- 1}{2x^{2}}\right) - ∫ \dfrac{1}{x}\left(\dfrac{- 1}{2x^{2}}\right)dx
(Using integration by parts)
=−lnx2x2+12(x−2−2)+C= - \dfrac{ln x}{2x^{2}} + \dfrac{1}{2}\left(\dfrac{x^{- 2}}{- 2}\right) + C
=−14x2(2lnx+1)+C= - \dfrac{1}{4x^{2}}\left(2ln x + 1\right) + C
⇒f(x)=−2lnx−1⇒ f\left(x\right) = - 2ln x - 1
∴f(e2)=−2ln⁡(e2)−1∴ f\left(e^{2}\right) = - 2\ln \left(e^{2}\right) - 1
=−4−1=−5= - 4 - 1 = - 5
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