please teach me how to integrate e^cos(t) sin(2t) dt from 0 to pi

To solve the integral \( \int_0^\pi e^{\cos(t)} \sin(2t) \, dt \), we can approach it using integration by parts or substitutions. In this case, it's simpler to use substitution and some known properties of integrals. Let’s break it down step by step.

### Step 1: Look for a potential substitution
We want to simplify the integral \( \int_0^\pi e^{\cos(t)} \sin(2t) \, dt \). We notice that the expression contains \( e^{\cos(t)} \) and \( \sin(2t) \). Let’s start by using a trigonometric identity for \( \sin(2t) \):

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

Thus, the integral becomes:

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

### Step 2: Use substitution
Let’s substitute \( u = \cos(t) \), which simplifies the expression considerably. This substitution works because \( du = -\sin(t) \, dt \), and we can also change the limits of integration accordingly:

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

Now, rewrite the integral in terms of \( u \):

\[
I = 2 \int_1^{-1} e^u \cdot (-du)
\]

The negative sign from \( du = -\sin(t) \, dt \) allows us to swap the limits of integration:

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

### Step 3: Solve the simplified integral
The integral of \( e^u \) is straightforward:

\[
\int e^u \, du = e^u
\]

Now, apply the limits of integration:

\[
I = 2 \left[ e^u \right]_{-1}^1 = 2 \left( e^1 - e^{-1} \right)
\]

### Step 4: Final answer
Thus, the value of the integral is:

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

This can be further simplified if needed, but the final answer is:

\[
\boxed{2 \left( e - \frac{1}{e} \right)}
\]

---

Would you like further details or clarification on any step? Here are some related questions:

1. How can you recognize when to use substitution in integrals?
2. Why did we use \( u = \cos(t) \) as the substitution here?
3. What are some common integrals involving \( e^x \) that are useful to know?
4. How does swapping the limits of integration affect the sign of the integral?
5. What are other methods for solving integrals involving trigonometric functions?

**Tip**: Substitution is a powerful technique when the integrand contains a function and its derivative, making the integral easier to solve.

Learn how to solve the integral of e^cos(t) * sin(2t) from 0 to pi using substitution and trigonometric identities. This detailed guide simplifies the process with a clear explanation, suitable for undergraduate students in calculus.

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