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

To integrate (e^x)*sin(x), 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. The formula for integration by parts is:

∫u dv = uv - ∫v du

where u and v are functions of x, and du/dx and dv/dx are their respective derivatives.

Let u = sin(x) and dv/dx = e^x. Then, du/dx = cos(x) and v = e^x.

Using the formula for integration by parts, we have:

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

Now, we need to integrate e^x*cos(x). Let u = cos(x) and dv/dx = e^x. Then, du/dx = -sin(x) and v = e^x.

Using the formula for integration by parts again, we have:

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

Substituting this back into the original equation, we get:

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

Simplifying, we get:

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

Dividing both sides by 2, we get:

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

where C is the constant of integration.