To solve the integral of \( \frac{1}{e^{7y}} \) with respect to \( y \), we can rewrite the expression to make it easier to integrate:

\[
\int \frac{1}{e^{7y}} \, dy = \int e^{-7y} \, dy
\]

Now, integrating \( e^{-7y} \) with respect to \( y \):

\[
\int e^{-7y} \, dy = \frac{e^{-7y}}{-7} = -\frac{1}{7} e^{-7y} + C
\]

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

### Final Answer:
\[
\int \frac{1}{e^{7y}} \, dy = -\frac{1}{7} e^{-7y} + C
\]

Would you like additional details or have any questions?

Here are some related questions:
1. How do you solve integrals involving exponential functions with different constants?
2. What happens when you integrate \( \frac{1}{e^{ky}} \) for a general constant \( k \)?
3. How do you differentiate \( -\frac{1}{7} e^{-7y} \) to check this result?
4. What are the rules for integrating exponential functions with negative exponents?
5. How does this integral change if the exponent is a function of \( y \), such as \( e^{f(y)} \)?

**Tip:** For integrals involving exponential terms, rewrite the term so that the exponent is easier to integrate or differentiate.

Learn how to find the integral of 1/e^7y with respect to y. This detailed solution covers integration techniques for exponential functions and uses the formula for integrating e^(kx) terms. Suitable for students in Grades 11-12.

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