$$ \int _ { 0 } ^ { \pi } e ^ { \cos ( t ) } \sin ( 2 t ) d t $$

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It seems that the integral $$ \int _ { 0 } ^ { \pi } e^{\cos(t)} \sin(2t) dt $$ cannot be directly computed using the available tools. However, we can approach this integral using integration by parts or other techniques.

### Step 1: Integration by Parts
We can use integration by parts, which states:
$$
\int u \, dv = uv - \int v \, du
$$
Let's choose:
- $$ u = \sin(2t) $$  (thus $$ du = 2\cos(2t) dt $$)
- $$ dv = e^{\cos(t)} dt $$ (we will need to find $$ v $$)

### Step 2: Finding $$ v $$
To find $$ v $$, we need to integrate $$ dv $$:
$$
v = \int e^{\cos(t)} dt
$$
This integral does not have a simple closed form, but we can denote it as $$ V(t) $$.

### Step 3: Applying Integration by Parts
Now we can apply integration by parts:
$$
\int e^{\cos(t)} \sin(2t) dt = V(t) \sin(2t) - \int V(t) \cdot 2\cos(2t) dt
$$

### Step 4: Evaluating the Limits
We will evaluate the expression from $$ 0 $$ to $$ \pi $$:
$$
\left[ V(t) \sin(2t) \right]_{0}^{\pi} - 2 \int_{0}^{\pi} V(t) \cos(2t) dt
$$

### Step 5: Evaluating $$ V(t) \sin(2t) $$ at the Limits
At $$ t = 0 $$:
$$
V(0) \sin(0) = 0
$$
At $$ t = \pi $$:
$$
V(\pi) \sin(2\pi) = 0
$$
Thus, the boundary terms vanish.

### Step 6: Final Integral
We are left with:
$$
- 2 \int_{0}^{\pi} V(t) \cos(2t) dt
$$
This integral still requires numerical methods or further techniques to evaluate $$ V(t) $$.

### Conclusion
The integral $$ \int _ { 0 } ^ { \pi } e^{\cos(t)} \sin(2t) dt $$ does not yield a simple closed form and may require numerical integration for an approximate value. Would you like to proceed with a numerical approximation?
The integral $$ \int _ { 0 } ^ { \pi } e^{\cos(t)} \sin(2t) dt $$ does not have a simple closed-form solution and may require numerical methods for an approximate value.

\( \int _ { 0 } ^ { \pi } e ^ { \cos ( t ) } \sin ( 2 t ) d t \)

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