Correct Answer - Option 3 :
\(\rm {(\ln 4)^2\over 2}\)
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
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 = \(\rm \int{\ln x\over x}dx\)
Let ln x = t ⇒\(\rm1\over x\)dx = dt
I = \(\rm \int{t}dt\)
I = \(\rm {t^2\over 2}\)
I = \(\rm {(\ln x)^2}\)
Putting the limits
I = \(\frac{1}{2} \times \rm \left[(\ln x)^2\right]_1^4\)
I = \(\frac{1}{2} \times \rm \left[(\ln 4)^2- (\ln 1)^2\right]\)
I = \(\frac{1}{2} \times \rm (\ln 4)^2 = {({ln \ 4})^2\over 2}
\)