1.

If y =  \(\rm \int{x\over1+x^4}\) dx and y(0) = 2, find the value of y at x = 2 1. tan-1 4 + 22. tan-1 2 + 23. \(1\over2\) tan-1 4 + 24. 2 tan-1 2 + 2

Answer» Correct Answer - Option 3 : \(1\over2\) tan-1 4 + 2

Concept:

Integral property:

  • ∫ xn dx = \(\rm x^{n+1}\over n+1\)+ C ; n ≠ -1
  • \(\rm∫ {1\over x} dx = \ln x\) + C
  • ∫ ex dx = ex+ C
  • ∫ ax dx = (ax/ln a) + C ; a > 0,  a ≠ 1
  • ∫ sin x dx = - cos x + C
  • ∫ cos x dx = sin x + C
  • ∫\(\rm {1\over1+x^2}\) dx = tan-1 + C

 

 

Calculation:

y = \(\rm ∫{x\over1+x^4}\) dx

⇒ y = \(\rm {1\over2}∫{2x\over1+x^4}\) dx

By substitution let x2 = t ⇒ 2x dx = dt

⇒ y = \(\rm {1\over2}∫{1\over1+t^2}\) dt

⇒ y = \(\rm {1\over2}\tan ^{-1}t\) + C

⇒ y = \(\rm {1\over2}\tan ^{-1}x^2\) + C

Given y(0) = 2

⇒ \(\rm {1\over2}\tan ^{-1}0\) + C = 2

⇒ C = 2

Now y = \(\rm {1\over2}\tan ^{-1}x^2\) + 2 

y(2) = \(\rm {1\over2}\tan ^{-1}2^2\) + 2

⇒ y(2) = \(\boldsymbol{\rm {1\over2}\tan ^{-1}4}\) + 2


Discussion

No Comment Found

Related InterviewSolutions