Question

$\int a^{x}dx=$

• A. $\frac{a^x}{x{\log }_{e}a}+C$
• B. $a^x{\log }_{e}a+C$
• C. $\frac{a^x}{{\log }_{e}a}+C$
• D. $\frac{x{a}^x}{{\log }_{e}a}+C$

Answer 1

The correct option is C

$\frac{a^x}{{\log }_{e}a}+C$

Explanation for Correct answer:

Solving the integral:

Let $a^x=e^{\log a^x}\left[\because e^{\log x}=x\right]$

Integrating on both sides,

$$\begin{gathered} \int a^{x}dx=\int e^{\log a^x}dx \\ =\int e^{x\log a}dx \\ \text{Let}u=x\log a\Rightarrow du=\log adx \\ =\frac{1}{\log a}\int e^u.du \\ =\frac{1}{\log a}{e}^u+C \\ =\frac{1}{\log a}{e}^{x\log a}+C \\ =\frac{1}{\log a}{e^{\log a}}^x+C\left[\because x\log a=\log a^x\right] \\ =\frac{1}{{\log }_{e}a}{a}^x+C \end{gathered}$$

Therefore, $\int a^{x}dx=$$\frac{a^x}{{\log }_{e}a}+C$

Hence, option (C) is the correct answer.