How to integrate e^x*sin(2x)?

To integrate e^x*sin(2x), use integration by parts.

Integration by parts is a technique used to integrate the product of two functions. It involves choosing one function to differentiate and the other to integrate. In this case, let u = sin(2x) and dv/dx = e^x. Then, du/dx = 2cos(2x) and v = e^x.

Using the formula for integration by parts, ∫u(dv/dx)dx = uv - ∫v(du/dx)dx, we have:

∫e^x*sin(2x)dx = e^x*sin(2x) - ∫e^x*2cos(2x)dx

To integrate ∫e^x*2cos(2x)dx, use integration by parts again. Let u = 2cos(2x) and dv/dx = e^x. Then, du/dx = -4sin(2x) and v = e^x.

Using the formula for integration by parts, we have:

∫e^x*2cos(2x)dx = 2e^x*cos(2x) + 4∫e^x*sin(2x)dx

Substituting this back into the original equation, we get:

∫e^x*sin(2x)dx = e^x*sin(2x) - 2e^x*cos(2x) - 4∫e^x*sin(2x)dx

Simplifying, we get:

5∫e^x*sin(2x)dx = e^x*sin(2x) - 2e^x*cos(2x)

Dividing both sides by 5, we get:

∫e^x*sin(2x)dx = (1/5)e^x*sin(2x) - (2/5)e^x*cos(2x) + C

Therefore, the integral of e^x*sin(2x) is (1/5)e^x*sin(2x) - (2/5)e^x*cos(2x) + C, where C is the constant of integration.