Let's solve the expression step by step:

Given: $-2\left( \cos(2t)e^t - 4\int -2\sin(2t)e^t dt \right)$

Step 1: Simplify the integral

We need to compute the integral: $\int -2\sin(2t)e^t dt$ This integral requires integration by parts. We can use the following formula for integration by parts: $\int u dv = uv - \int v du$ Let $u = \sin(2t)$ and $dv = -2e^t dt$.

1. Differentiate $u$ and integrate $dv$:

• $du = 2\cos(2t) dt$
• $v = -2e^t$

2. Applying the integration by parts formula:

$\int -2\sin(2t)e^t dt = (-2e^t)(\sin(2t)) - \int -2e^t \cdot 2\cos(2t) dt$ $= -2e^t\sin(2t) + 4\int e^t\cos(2t) dt$

Step 2: Solve the remaining integral

Now, we need to compute: $\int e^t \cos(2t) dt$ We will again use integration by parts for this integral. Let:

• $u = \cos(2t)$ and $dv = e^t dt$

1. Differentiate $u$ and integrate $dv$:

• $du = -2\sin(2t) dt$
• $v = e^t$

2. Applying the integration by parts formula:

[ \int e^t \