Integration of e^(3x)sinx dx

To integrate this expression, we will use integration by parts.

First, we need to choose our u and dv. Let u = sinx and dv = e^(3x)dx.

Now, we can find du and v by taking the derivative of u and integrating dv, respectively.

du/dx = cosx v = (1/3)e^(3x)

Using the integration by parts formula:

∫ u dv = uv - ∫ v du

we can now substitute in our values:

∫ e^(3x)sinx dx = (-1/3)e^(3x)cosx + (1/3)∫ e^(3x)cosx dx

Now, we can apply integration by parts again by letting u = cosx and dv = e^(3x)dx.

du/dx = -sinx v = (1/3)e^(3x)

Substituting these values into our formula gives us:

∫ e^(3x)sinx dx = (-1/3)e^(3x)cosx + (1/9)e^(3x)sinx - (1/9)∫ e^(3x)sinx dx

Moving the last term to the left side and simplifying:

(10/27)∫ e^(3x)sinx dx = (-1/3)e^(3x)cosx + (1/9)e^(3x)sinx

Finally, we can solve for our integral:

∫ e^(3x)sinx dx = (27/10) * (-1/3)e^(3x)cosx + (27/10) * (1/9)e^(3x)sinx

Therefore,

∫ e^(3x)sinx dx = (3/10)e^(3x)(sinx - 3cosx) + C

where C is the constant of integration.