Evaluate the integral using method of integration by parts:

(integral sign)(e^(2x))sin(5x)dx

You can do the partial integration by integrating the e^(2x) term. Then you end up with an integral proportional to e^(2x)cos(5x). You then integrate that again using partial integration, again integrating the e^(2x) term. You then get the integral you started out with back plus some other terms. So, if we call the integral I, you have a relation of the form:

I = some terms - a I

where a is some constant. Then you solve for I:

I = (some terms)/(1+a)

If you are familiar with complex numbers, then you can write:

sin(5x) = Im[exp(5 i x)

The integral is thus the imaginary part of the integral of exp[(2 + 5 i)x]. This is:

Im {exp[(2 + 5 i)x]/(2 + 5 i)} =

1/29 exp(2x) [2 sin(5 x) -5 cos(5x)]

To evaluate the given integral ∫ e^(2x) sin(5x) dx using the method of integration by parts, we will use the formula:

∫ u * dv = u * v - ∫ v * du

Step 1: Choose u and dv
Let's assign u = sin(5x) and dv = e^(2x) dx.

Step 2: Differentiate u to find du
To find du, we differentiate u = sin(5x) with respect to x.
du = (d/dx) (sin(5x)) dx

The derivative of sin(5x) with respect to x can be found using the chain rule. As per the chain rule, the derivative of sin(mx) with respect to x is cos(mx) * m. Therefore,
du = (d/dx) (sin(5x)) dx = cos(5x) * 5 dx = 5 cos(5x) dx

Step 3: Integrate dv to find v
To find v, we integrate dv = e^(2x) dx.
∫ e^(2x) dx = (1/2) e^(2x)

Step 4: Apply the formula
Substituting the values of u, v, du, and dv into the formula, we get:
∫ e^(2x) sin(5x) dx = u * v - ∫ v * du
∫ e^(2x) sin(5x) dx = sin(5x) * (1/2) e^(2x) - ∫ (1/2) e^(2x) * 5 cos(5x) dx

Simplifying further:
∫ e^(2x) sin(5x) dx = (1/2) sin(5x) e^(2x) - (5/2) ∫ e^(2x) cos(5x) dx

Now, we need to integrate the remaining term ∫ e^(2x) cos(5x) dx. This can be done using integration by parts again.

The process will continue iteratively until we reach an integral that can be solved easily or express it as another form based on special cases or identities.