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The given integral is:

\[
I = \int_0^{\pi} e^{\cos(t)} \sin(2t) \, dt
\]

### Step 1: Substitution

Let's first make a substitution. We can start by letting \( u = \cos(t) \). Then,

\[
du = -\sin(t) \, dt
\]

However, this doesn't directly simplify the given integral because we have a \( \sin(2t) \) term instead of just \( \sin(t) \). We need to handle this \( \sin(2t) \) term. Using the identity:

\[
\sin(2t) = 2\sin(t)\cos(t)
\]

### Step 2: Adjust the integral using this identity

Thus, the integral becomes:

\[
I = \int_0^{\pi} e^{\cos(t)} 2\sin(t)\cos(t) \, dt
\]

We can now make the substitution \( u = \cos(t) \), where \( du = -\sin(t) \, dt \). This transforms the integral into:

\[
I = -2 \int_{1}^{-1} e^{u} u \, du
\]

Notice that the limits of integration change because \( \cos(0) = 1 \) and \( \cos(\pi) = -1 \).

### Step 3: Reverse limits of integration

To make the integral easier to evaluate, we reverse the limits of integration, which introduces a negative sign:

\[
I = 2 \int_{-1}^{1} e^u u \, du
\]

### Step 4: Integration by parts

Now, we apply integration by parts. Let:

\[
v = u \quad \text{and} \quad dw = e^u \, du
\]

Thus, we have:

\[
dv = du \quad \text{and} \quad w = e^u
\]

Using the integration by parts formula \( \int v \, dw = vw - \int w \, dv \), we get:

\[
I = 2 \left[ u e^u \bigg|_{-1}^{1} - \int_{-1}^{1} e^u \, du \right]
\]

### Step 5: Evaluate the terms

First, evaluate \( u e^u \) at the limits:

\[
u e^u \bigg|_{-1}^{1} = (1 \cdot e^1) - (-1 \cdot e^{-1}) = e + \frac{1}{e}
\]

Now, evaluate the remaining integral:

\[
\int_{-1}^{1} e^u \, du = e^u \bigg|_{-1}^{1} = e - \frac{1}{e}
\]

### Step 6: Final result

Now substitute everything back:

\[
I = 2 \left[ \left(e + \frac{1}{e}\right) - \left(e - \frac{1}{e}\right) \right]
\]

Simplifying:

\[
I = 2 \left( \frac{2}{e} \right) = \frac{4}{e}
\]

Thus, the value of the integral is:

\[
I = \frac{4}{e}
\]

Would you like further details on any step, or do you have any questions?

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Here are 5 related questions:

1. What are the key conditions to use integration by parts?
2. How can different substitutions simplify integration problems?
3. Why does reversing the limits of integration introduce a negative sign?
4. What are some common identities involving trigonometric functions used in integrals?
5. How do exponential functions behave under integration?

**Tip:** When applying integration by parts, choose \( u \) and \( dv \) strategically to simplify the remaining integral.

Evaluate the integral \( \int_0^{\pi} e^{\cos(t)} \sin(2t) \, dt \) using substitution and integration by parts.

Learn how to evaluate the integral \( \int_0^{\pi} e^{\cos(t)} \sin(2t) \, dt \) by making a substitution and applying the integration by parts method. This solution includes detailed steps using trigonometric identities and integration techniques, making it suitable for undergraduate calculus students.

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