- #1
Emmanuel_Euler
- 142
- 11
what is the integral of e^-(x/2) * sin(a*x) dx??
The integral of e^-(x/2) * sin(a*x) dx is (-2/a) * e^-(x/2) * cos(a*x) + C, where C is the constant of integration.
To solve this integral, we can use the integration by parts method. Let u = e^-(x/2) and dv = sin(a*x) dx. Then, du = (-1/2) * e^-(x/2) and v = (-1/a) * cos(a*x). Plugging these into the integration by parts formula, we get (-2/a) * e^-(x/2) * cos(a*x) - (-1/2a) * integral of e^-(x/2) * cos(a*x) dx. Solving for the integral, we get (-2/a) * e^-(x/2) * cos(a*x) + C.
e^-(x/2) * sin(a*x) is a common integrand in many mathematical models, including those related to oscillatory systems and damped harmonic motion. Its integral can be used to calculate the total displacement or energy in these systems.
Yes, the integral can also be solved using substitution. Let u = -(x/2) and du = (-1/2) * dx. Then, the integral becomes (-2/a) * e^u * sin(a*x) du. We can then use the trigonometric identity sin(a*x) = (1/2i) * (e^(iax) - e^(-iax)) to rewrite the integral as (-1/ia) * (e^(iax) - e^(-iax)) * e^u du. Combining like terms and integrating, we get (-2/a) * e^-(x/2) * cos(a*x) + C.
Yes, there is a general formula for the integral of e^-(x/2) * sin(a*x) dx. It is (-2/a) * e^-(x/2) * cos(a*x) + C, where C is the constant of integration. This formula can be derived using various integration techniques, such as integration by parts or substitution.