∫e sin(x) cos(x)dx.

Sure, let's solve the integral step by step.

First, we can use the product-to-sum identities from trigonometry to simplify the integral. The identity we need is:

sin(x)cos(x) = 1/2sin(2x)

So, the integral becomes:

∫e sin(x) cos(x) dx = ∫e * 1/2sin(2x) dx = 1/2 ∫e sin(2x) dx

Now, we can use the method of integration by parts. The formula for integration by parts is:

∫udv = uv - ∫vdu

We let u = e and dv = sin(2x) dx. Then du = e dx and v = -1/2 cos(2x).

Substituting these into the integration by parts formula gives:

1/2 ∫e sin(2x) dx = 1/2 * e * -1/2 cos(2x) - ∫-1/2 cos(2x) * e dx = -1/4 e cos(2x) + 1/2 ∫e cos(2x) dx

Now, we need to integrate ∫e cos(2x) dx. We can use integration by parts again, this time letting u = e and dv = cos(2x) dx. Then du = e dx and v = 1/2 sin(2x).

Substituting these into the integration by parts formula gives:

1/2 ∫e cos(2x) dx = 1/2 * e * 1/2 sin(2x) - ∫1/2 sin(2x) * e dx = 1/4 e sin(2x) - 1/2 ∫e sin(2x) dx

Substituting this back into the original integral gives:

-1/4 e cos(2x) + 1/2 (1/4 e sin(2x) - 1/2 ∫e sin(2x) dx)

Solving this equation for the original integral gives:

∫e sin(x) cos(x) dx = -1/4 e cos(2x) + 1/8 e sin(2x) + C

where C is the constant of integration.