To solve for the constant \( C \), we need to evaluate the indefinite integral and then substitute a known value of \( x \) and \( y \) to find \( C \). However, since the problem involves an indefinite integral and we don't have specific values for \( x \) and \( y \), we need to assume a solution method for the integral.

First, let's rewrite the integral:

\[
\int \frac{dy}{e^{4y} + 1}
\]

This integral doesn't have a simple antiderivative in terms of elementary functions. However, let's assume that the integral is given in this form, and we want to match it with the equation \( 2\sqrt{x} + C \).

Since the equation is:

\[
\int \frac{dy}{e^{4y} + 1} = 2\sqrt{x} + C
\]

Given that the integral is already evaluated, and the right-hand side is in terms of \( x \), we don't need to compute the integral directly. Instead, we need to find \( C \) by assuming an initial condition or specific values of \( x \) and \( y \).

Let's assume \( x = 0 \) and \( y = 0 \):

\[
\int \frac{dy}{e^{4(0)} + 1} = 2\sqrt{0} + C
\]

\[
\int \frac{dy}{2} = C
\]

The integral \( \int \frac{dy}{2} \) is simply \( \frac{y}{2} \), so:

\[
\frac{y}{2} = C
\]

Since \( y = 0 \) at \( x = 0 \), we get:

\[
C = 0
\]

Thus, under the assumption that \( y = 0 \) when \( x = 0 \), the constant \( C \) is 0.

Would you like further details or have any questions?

Here are 5 related questions:
1. What are the different methods for solving integrals with complex functions?
2. How does the method of substitution help in solving integrals?
3. What is the importance of boundary conditions in solving indefinite integrals?
4. How do definite and indefinite integrals differ in their solutions?
5. Can all integrals be solved using elementary functions?

**Tip:** When solving for constants in integration problems, always consider applying initial or boundary conditions to determine the specific value of the constant.

Learn how to solve the integral \( \int \frac{dy}{e^{4y} + 1} = 2\sqrt{x} + C \) with a step-by-step solution method. Understand the application of initial conditions to determine the constant \( C \). Suitable for advanced college-level mathematics.

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