Correct Answer - Option 4 :
\(\rm {1\over2}\ln (x^4 + 4x^2 + 5)+c\)
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
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 be determined.
- Substitution is done and the integral function is then integrated.
- Finally, the initial variable t, to be returned.
Calculation:
I = \(\rm \int {2x^3+4x\over x^4+5+4x^2}dx\)
I = \(\rm \int {2x^3+4x\over x^4+4x^2 +5}dx\)
I = \(\rm \frac 1 2 \int {4x^3+8x\over x^4+4x^2 +5}dx\)
Let x4 + 4x2 = t
⇒ (4x3 + 8x)dx = dt
Now,
I = \(\rm {1\over2}\int {1\over t}dt\)
I = \(\rm {1\over2}\ln t + c\)
I = \(\boldsymbol{\rm {1\over2}\ln (x^4 + 4x^2 + 5)+c}\)