Question

$\int \log_e (a^x + x^a) dx$ is equal to

• A. $\frac{x^{a+1}}{a+1} + c$
• B. $\frac{x^{a+1}}{a+1} + a^x \log a + c$
• C. $x^{a+1} + c$
• D. $\frac{x^{a+1}}{a+1} + a^x \frac{1}{\log a} + c$

Hint:

$\int a^x dx = \frac{a^x}{\ln a} + c$, $\int x^n dx = \frac{x^{n+1}}{n+1} + c$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The integral $\int \log(a^x + x^a) dx$ does not have a simple elementary form. The options suggest it's the integral of the sum, not the log. So probably the question is $\int (a^x + x^a) dx$.
Step 2: Detailed Explanation:
If the integrand is $a^x + x^a$, then $\int a^x dx = \frac{a^x}{\log a} + c$, and $\int x^a dx = \frac{x^{a+1}}{a+1} + c$. So the integral is $\frac{x^{a+1}}{a+1} + \frac{a^x}{\log a} + c$, which matches option (D).
Step 3: Final Answer:
The integral is $\frac{x^{a+1}}{a+1} + \frac{a^x}{\log a} + c$.