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?

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.