Find the indefinite integral. (Remember the constant of integration.) \[ \int \frac{e^{2 x}}{49+e^{4 x}} d x \]

Calculate the integral \( \int \frac{e^{2x}}{49+e^{4x}} dx \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \frac{e^{2x}}{49+e^{4x}} dx\) - step1: Use the substitution \(dx=\frac{1}{2e^{2x}} dt\) to transform the integral\(:\) \(\int \frac{e^{2x}}{49+e^{4x}}\times \frac{1}{2e^{2x}} dt\) - step2: Simplify: \(\int \frac{1}{98+2\left(e^{2x}\right)^{2}} dt\) - step3: Use the substitution \(t=e^{2x}\) to transform the integral\(:\) \(\int \frac{1}{98+2t^{2}} dt\) - step4: Rewrite the expression: \(\int \frac{1}{2}\times \frac{1}{49+t^{2}} dt\) - step5: Use properties of integrals: \(\frac{1}{2}\times \int \frac{1}{49+t^{2}} dt\) - step6: Evaluate the integral: \(\frac{1}{2}\times \frac{1}{7}\arctan\left(\frac{t}{7}\right)\) - step7: Multiply the terms: \(\frac{1}{14}\arctan\left(\frac{t}{7}\right)\) - step8: Substitute back: \(\frac{1}{14}\arctan\left(\frac{e^{2x}}{7}\right)\) - step9: Add the constant of integral C: \(\frac{1}{14}\arctan\left(\frac{e^{2x}}{7}\right) + C, C \in \mathbb{R}\) The indefinite integral of \( \frac{e^{2x}}{49+e^{4x}} \) with respect to \( x \) is \( \frac{1}{14}\arctan\left(\frac{e^{2x}}{7}\right) + C \), where \( C \) is the constant of integration.