Correct Answer - Option 4 :
\(\rm \frac{x^5}{5}+c\)
Concept:
- eln x = x.
- ∫ xn dx = \(\rm \frac{x^{n+1}}{n+1}+c\).
Calculation:
Let I = \(\rm \int e^{\left(2\ln x+\ln x^2\right)}dx\)
⇒ I = \(\rm \int e^{\left(\ln x^2+\ln x^2\right)}dx\)
⇒ I = \(\rm \int e^{2\ln x^2}dx\)
⇒ I = \(\rm \int e^{\ln x^4}dx\)
⇒ I = ∫ x4 dx
∴ The value of the integral \(\rm \int e^{\left(2\ln x+\ln x^2\right)}dx\) is \(\rm \frac{x^5}{5}+c\).
