5. Evaluate the indefinite integral. (Use \( C \) for the constant of integration.) \[ \int \frac{\sin (2 x)}{44+\cos ^{2}(x)} d x \]

It seems that the integral I attempted to evaluate is not supported directly. However, I can guide you through the process of solving it step by step. To evaluate the integral \[ \int \frac{\sin(2x)}{44 + \cos^2(x)} \, dx, \] we can use the following steps: 1. **Use the double angle identity for sine**: Recall that \(\sin(2x) = 2\sin(x)\cos(x)\). Thus, we can rewrite the integral as: \[ \int \frac{2\sin(x)\cos(x)}{44 + \cos^2(x)} \, dx. \] 2. **Substitution**: Let \(u = \cos(x)\). Then, \(du = -\sin(x) \, dx\) or \(-du = \sin(x) \, dx\). The integral becomes: \[ -2 \int \frac{u}{44 + u^2} \, du. \] 3. **Simplifying the integral**: We can now evaluate the integral: \[ -2 \int \frac{u}{44 + u^2} \, du. \] 4. **Using a standard integral formula**: The integral \(\int \frac{u}{a + u^2} \, du\) can be solved using the formula: \[ \int \frac{u}{a + u^2} \, du = \frac{1}{2} \ln |a + u^2| + C. \] 5. **Applying the formula**: In our case, \(a = 44\): \[ -2 \left( \frac{1}{2} \ln |44 + u^2| \right) + C = -\ln |44 + u^2| + C. \] 6. **Substituting back**: Recall that \(u = \cos(x)\): \[ -\ln |44 + \cos^2(x)| + C. \] Thus, the final result for the indefinite integral is: \[ \int \frac{\sin(2x)}{44 + \cos^2(x)} \, dx = -\ln |44 + \cos^2(x)| + C. \]