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

NTA Abhyas 2020 · Mathematics · Indefinite Integration

Let f(x) = 2x + 1 and g(x) = ∫ f(x) x^ 2 (x + 1) ^ 2 dx . If 6g(2)...

Let f(x)=2x+1andf\left(x\right) = 2x + 1 andg\left(x\right) = ∫ \dfrac{f\left(x\right)}{x^{2}{ \left(x + 1\right)}^{2}}dx.If6g(2). If 6g\left(2\right)+ 1 = 0,theng(−12)then g\left(- \dfrac{1}{2}\right)is equal to

Options

  1. A44
  2. B−4- 4
  3. C33
  4. D22

Correct answer

A. 44

Step-by-step solution

g(x)=∫(x+1)2−x2x2(x+1)2dx=∫(1x2−1(x+1)2)dx=−1x+1x+1+cg\left(x\right) = ∫ \dfrac{\left(x + 1\right)^{2} - x^{2}}{x^{2}\left(x + 1\right)^{2}}dx = ∫ \left(\dfrac{1}{x2} - \dfrac{1}{\left(x + 1\right)^{2}}\right)dx = - \dfrac{1}{x} + \dfrac{1}{x + 1} + c
=−1x(x+1)+c= - \dfrac{1}{x\left(x + 1\right)} + c
g(2)=−16+c=−16⇒c=0g\left(2\right) = - \dfrac{1}{6} + c = - \dfrac{1}{6} ⇒ c = 0
Hence, g(−12)=4g\left(- \dfrac{1}{2}\right) = 4
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