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First make a substitution and then use integration by parts to evaluate the integral.

The integral provided is:

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

We are asked to make a substitution and then use integration by parts to evaluate the integral. Let's follow these steps.

### Step 1: Substitution

Consider the substitution:

\[
u = \cos t
\]
Then, the derivative is:
\[
du = -\sin t \, dt
\]

From the original integral, we have \(\sin(2t)\), which we can rewrite using a trigonometric identity:

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

So, the integral becomes:

\[
\int_0^\pi e^{\cos t} \cdot 2 \sin t \cos t \, dt
\]

Now, substitute \(u = \cos t\), which implies \(du = -\sin t \, dt\). The limits of integration also change:

- When \(t = 0\), \(u = \cos(0) = 1\).
- When \(t = \pi\), \(u = \cos(\pi) = -1\).

The integral now transforms to:

\[
-2 \int_1^{-1} e^u u \, du
\]

### Step 2: Integration by Parts

We apply integration by parts to solve the integral. Recall the formula:

\[
\int u \, dv = uv - \int v \, du
\]

Let:
- \(v = e^u\)
- \(du = u \, du\)

Now, applying integration by parts:

The evaluated integral is:

\[
\int_0^\pi e^{\cos t} \sin(2t) \, dt = 4e^{-1}
\]

This means the final result of the integral is \(4e^{-1}\), or approximately \(4 \times 0.3679 = 1.4716\).

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

Here are 5 related questions for practice:
1. How would the result change if the limits of integration were from 0 to \(\frac{\pi}{2}\)?
2. What if the integrand was \(e^{\sin t} \sin(2t)\)?
3. Can you solve the integral using a different method besides integration by parts?
4. How does changing the power of the sine function in the integrand affect the result?
5. What is the result if you replace \(\sin(2t)\) with \(\cos(2t)\)?

**Tip:** When using integration by parts, carefully choose the parts so that the resulting integrals simplify effectively.

Explore how to evaluate the integral \(\int_0^\pi e^{\cos t} \sin(2t) \, dt\) using substitution and integration by parts. This problem involves key calculus concepts, including integration by parts and trigonometric identities, suitable for college-level students.

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