Step 1: Understanding the Concept:
The integral involves an exponential function multiplied by trigonometric functions. The first step is to simplify the trigonometric part using double-angle identities to make integration easier.
Step 2: Key Formula or Approach:
Use the trigonometric identity: $\sin 2x = 2 \sin x \cos x \implies \sin x \cos x = \frac{1}{2} \sin 2x$.
Then apply the standard integration formula:
\[ \int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin bx - b \cos bx) + C \]
Step 3: Detailed Explanation:
Let the integral be $I$:
\[ I = \int e^x \sin x \cos x \, dx \]
Substitute the double-angle identity:
\[ I = \int e^x \left( \frac{\sin 2x}{2} \right) \, dx = \frac{1}{2} \int e^x \sin 2x \, dx \]
Now, compare $\int e^x \sin 2x \, dx$ with the standard form $\int e^{ax} \sin bx \, dx$:
Here, $a = 1$ and $b = 2$.
Applying the formula:
\[ \int e^x \sin 2x \, dx = \frac{e^{1 \cdot x}}{1^2 + 2^2} (1 \cdot \sin 2x - 2 \cdot \cos 2x) \]
\[ = \frac{e^x}{1 + 4} (\sin 2x - 2 \cos 2x) = \frac{e^x}{5} (\sin 2x - 2 \cos 2x) \]
Now, substitute this result back into $I$:
\[ I = \frac{1}{2} \left[ \frac{e^x}{5} (\sin 2x - 2 \cos 2x) \right] + C \]
\[ I = \frac{e^x}{10} (\sin 2x - 2 \cos 2x) + C \]
Step 4: Final Answer:
The evaluated integral is $\frac{e^x}{10} (\sin 2x - 2 \cos 2x) + C$.