e^x/3 dx The answer is 3e^x/3 + C explain why

The integral you're evaluating is:

\[
\int \frac{e^x}{3} \, dx
\]

To explain why the answer is \( 3e^{x/3} + C \), let's break it down step by step.

### Step 1: Simplify the integral
The given integral can be written as:

\[
\int \frac{1}{3} e^x \, dx
\]

Since \( \frac{1}{3} \) is a constant, you can factor it out of the integral:

\[
\frac{1}{3} \int e^x \, dx
\]

### Step 2: Integrate \( e^x \)
The integral of \( e^x \) with respect to \( x \) is simply \( e^x \), because the exponential function \( e^x \) is its own derivative:

\[
\int e^x \, dx = e^x + C_1
\]

where \( C_1 \) is the constant of integration.

### Step 3: Multiply by the constant
Now, multiply the result of the integral by the constant \( \frac{1}{3} \):

\[
\frac{1}{3} \cdot (e^x + C_1) = \frac{1}{3} e^x + \frac{1}{3} C_1
\]

### Step 4: Simplify the constant
The constant term \( \frac{1}{3} C_1 \) can be simplified to just \( C \), since any constant times a constant is still a constant. So, we can rewrite the answer as:

\[
\frac{1}{3} e^x + C
\]

This is the result of the integral \( \frac{e^x}{3} dx \).

However, it seems there may have been a misunderstanding in interpreting the integral you provided as leading to \( 3e^{x/3} + C \). If the original problem had been \( \int e^{x/3} \, dx \), we would approach it differently, using substitution. Let me know if this might have been the case, and I can clarify further.

### Do you have any questions or need further clarification on this explanation?

Here are some related questions that might interest you:
1. What is the integral of \( e^{kx} \) where \( k \) is a constant?
2. How does substitution work in integration, especially for functions like \( e^{x/3} \)?
3. Why do constants of integration combine into a single constant?
4. What are the general rules for integrating exponential functions?
5. How does integration by parts apply to more complex functions involving exponentials?

**Tip:** Always watch for constants in integrals, as they can factor in and simplify the problem early on!

The query submitted by the user: e^x/3 dx. The answer is 3e^x/3 + C. Explain why.

Understand why the integral of e^x/3 dx results in 3e^x/3 + C with a detailed step-by-step explanation. This solution covers key concepts like exponential functions and integration techniques, suitable for high school students in Grades 10-12.

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