1. **State the problem:** We want to solve for the integral $$\int e^{4x} \sin(5x) \, dx$$ given the system: $$\int e^{4x} \sin(5x) \, dx = -\frac{1}{5} e^{4x} \cos(5x) + \frac{4}{5} \int e^{4x} \cos(5x) \, dx$$ and $$\int e^{4x} \cos(5x) \, dx = \frac{1}{5} e^{4x} \sin(5x) - \frac{4}{5} \int e^{4x} \sin(5x) \, dx$$ 2. **Substitute the second integral into the first:** $$\int e^{4x} \sin(5x) \, dx = -\frac{1}{5} e^{4x} \cos(5x) + \frac{4}{5} \left( \frac{1}{5} e^{4x} \sin(5x) - \frac{4}{5} \int e^{4x} \sin(5x) \, dx \right)$$ 3. **Distribute the $ \frac{4}{5} $:** $$\int e^{4x} \sin(5x) \, dx = -\frac{1}{5} e^{4x} \cos(5x) + \frac{4}{25} e^{4x} \sin(5x) - \frac{16}{25} \int e^{4x} \sin(5x) \, dx$$ 4. **Group the integral terms on one side:** $$\int e^{4x} \sin(5x) \, dx + \frac{16}{25} \int e^{4x} \sin(5x) \, dx = -\frac{1}{5} e^{4x} \cos(5x) + \frac{4}{25} e^{4x} \sin(5x)$$ 5. **Combine like terms:** $$\left(1 + \frac{16}{25}\right) \int e^{4x} \sin(5x) \, dx = -\frac{1}{5} e^{4x} \cos(5x) + \frac{4}{25} e^{4x} \sin(5x)$$ 6. **Simplify the coefficient:** $$\frac{25}{25} + \frac{16}{25} = \frac{41}{25}$$ So, $$\frac{41}{25} \int e^{4x} \sin(5x) \, dx = -\frac{1}{5} e^{4x} \cos(5x) + \frac{4}{25} e^{4x} \sin(5x)$$ 7. **Solve for the integral:** $$\int e^{4x} \sin(5x) \, dx = \frac{25}{41} \left(-\frac{1}{5} e^{4x} \cos(5x) + \frac{4}{25} e^{4x} \sin(5x)\right) + C$$ 8. **Distribute $ \frac{25}{41} $:** $$\int e^{4x} \sin(5x) \, dx = -\frac{25}{41} \cdot \frac{1}{5} e^{4x} \cos(5x) + \frac{25}{41} \cdot \frac{4}{25} e^{4x} \sin(5x) + C$$ 9. **Simplify the fractions:** $$-\frac{25}{41} \cdot \frac{1}{5} = -\frac{5}{41}$$ $$\frac{25}{41} \cdot \frac{4}{25} = \frac{4}{41}$$ 10. **Final answer:** $$\boxed{\int e^{4x} \sin(5x) \, dx = -\frac{5}{41} e^{4x} \cos(5x) + \frac{4}{41} e^{4x} \sin(5x) + C}$$