Correct Answer - Option 1 :
\(\rm -{1\over2x^2+1}+c\)
Concept:
Integral property:
- ∫ xn dx = \(\rm x^{n+1}\over n+1\)+ C ; n ≠ -1
-
\(\rm\int {1\over x} dx = \ln x\) + 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
Substitution method: If the function cannot be integrated directly substitution method is used. To integration by substitution is used in the following steps:
- A new variable is to be chosen, say “t”
- The value of dt is to is to be determined.
- Substitution is done and integral function is then integrated.
- Finally, initial variable t, to be returned.
Calculation:
I = \(\rm \int {4x\over(2x^2+1)^2}\)
Let 2x2 + 1 = t
Differentiating both sides
⇒ 4x dx = dt
Substituting 2x2 + 1 by t and dx = dt/4x
⇒ I = \(\rm \int {1\over(t)^2}\) dt
⇒ I = \(\rm \left[ {(t)^{-1}\over-1}\right] + c\)
⇒ I = \(\rm {-1\over t}\) + c
Substituting t as x2
⇒ I = \(\boldsymbol{\rm -{1\over2x^2+1}+c}\)
