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

NTA Abhyas 2020 · Mathematics · Indefinite Integration

If f (x) is a differentiable function such that ∫ f(x)dx = 2...

If ff\left(x\right)isadifferentiablefunctionsuchthatis a differentiable function such that∫ f\left(x\right)dx = 2\lleft[f(x)]2{\l left[f\left(x\right)]}^{2} + C,(where,, (where,C is the constant of integration)andf(1)=1/4,and f \left(1\right) = 1 / 4, then f(π)f\left(\pi\right)equals

Options

  1. Aπ\pi
  2. Bπ4\dfrac{π}{4}
  3. C23\dfrac{2}{3}
  4. Dπ6\dfrac{\pi}{6}

Correct answer

B. π4\dfrac{π}{4}

Step-by-step solution

Differentiating w.r.t. ‘x’‘x’ we get f(x)=4ff \left(x\right) = 4f\left(x\right)\cdotf^{′} \left(x\right)
f′(x)=0f^{′} \left(x\right)= 0 or 14=\dfrac{1}{4} = f \left(x\right)<br/>Integrating,wehave<br/><br />Integrating, we have<br />∫ \dfrac{1}{4} dx=∫f′(x)x = ∫ f^{′} \left(x\right) dx<br/><br />⇒ x4=f(x)+C\dfrac{x}{4} =f\left(x\right)+ C
As f(1)=14\toC=0f\left(1\right) =\dfrac{1}{4} \toC =0
⇒f(x)=x4⇒ f \left(x\right) =\dfrac{x}{4}
⇒f(π)=π4⇒ f \left(\pi\right) =\dfrac{\pi}{4}
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