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 \

Explore the solution to -2(cos(2t)e^t - 4∫-2sin(2t)e^t dt) using integration by parts. Learn how to handle trigonometric functions like sin(2t) and cos(2t) combined with the exponential function e^t. Suitable for Grades 11-12 and early university students.

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