Question

$\int e^{x\log a}{e}^{x}dx=$

• A. $\left(\frac{a^x}{\log ae}\right)+c$
• B. $\left(\frac{e^x}{1+{\log }_{e}a}\right)+c$
• C. ${\left(ae\right)}^x+c$
• D. $\left(\frac{a{e}^x}{{\log }_{e}ae}\right)+c$
• E. $\left(\frac{a^x{e}^x}{{\log }_{x}e}\right)+c$

Answer 1

The correct option is D

$\left(\frac{a{e}^x}{{\log }_{e}ae}\right)+c$

Explanation for the correct option:

Finding the given integral:

$$\begin{gathered} \int e^{x\log a}{e}^{x}dx \\ \text{Let's take}{a}^x=e^{x\log a} \\ =\int a^x\cdot e^{x}dx \\ =\int {\left(ae\right)}^{x}dx \\ =\frac{{\left(ae\right)}^x}{{\log }_{e}ae}+c\left[\because \int a^{x}dx=\frac{a^x}{\log a}+c\right] \end{gathered}$$

Hence, the correct answer is option (D).