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

To integrate e^x*sin(x), use integration by parts with u = sin(x) and dv/dx = e^x.

To integrate e^x*sin(x), we can use integration by parts. 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 can use integration by parts again with u = cos(x) and dv/dx = e^x. Then, du/dx = -sin(x) and v = e^x.

Plugging these values into the formula for integration by parts, we get:

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

Rearranging, we get:

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

Dividing both sides by 2, we get:

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

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

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