To solve the integral $\int \frac{e^x}{3} \, dx$, using the substitution $u = \frac{x}{3}$, follow these steps:

Step 1: Apply substitution

Given $u = \frac{x}{3}$, differentiate both sides with respect to $x$: $du = \frac{1}{3} \, dx \quad \text{or} \quad dx = 3 \, du$

Step 2: Substitute in the integral

Substitute $u = \frac{x}{3}$ and $dx = 3 \, du$ into the integral: $\int \frac{e^x}{3} \, dx = \int \frac{e^{3u}}{3} \cdot 3 \, du = \int e^u \, du$

Step 3: Solve the new integral

The integral of $e^u$ with respect to $u$ is straightforward: $\int e^u \, du = e^u + C$

Step 4: Substitute $u = \frac{x}{3}$ back into the solution

Now, replace $u$ with $\frac{x}{3}$ to return to the original variable: $e^u + C = e^{\frac{x}{3}} + C$

Final Answer:

$\int \frac{e^x}{3} \, dx = e^{\frac{x}{3}} + C$

Would you like further details on any step?

5 Related Questions:

1. How do we perform substitution for integrals?
2. Can we apply substitution in trigonometric integrals?
3. What are the common methods of solving exponential integrals?
4. How does chain rule relate to substitution in integrals?
5. How do you handle more complex exponential integrals?

Tip:

Always verify your substitution by differentiating the result to ensure it matches the original integrand.