Question

\(\int_{0}^{\frac{\pi}{2}} \sin 2x\,e^{\sin x}\,dx\) is equal to

• A. \(4\)
• B. \(3\)
• C. \(2\)
• D. \(1\)
• E. \(0\)

Hint:

For integrals involving \(\sin x\) and \(e^{\sin x}\), substitution \(t=\sin x\) simplifies the problem significantly.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires evaluating a definite integral that involves a product of trigonometric and exponential functions. The structure suggests that a combination of trigonometric identities and integration by substitution, followed by integration by parts, will be necessary.
Step 2: Key Formula or Approach:
1. Use the double angle identity \( \sin 2x = 2 \sin x \cos x \).
2. Use substitution, letting \( u \) be a part of the exponent. A good choice is \( u = \sin x \).
3. The resulting integral will likely require integration by parts: \( \int u dv = uv - \int v du \).
Step 3: Detailed Explanation:
First, rewrite the integrand using the double angle identity:
\[ I = \int_0^{\pi/2} (2 \sin x \cos x) e^{\sin x} dx \] Now, let's use the substitution \( u = \sin x \).
Then \( du = \cos x dx \).
We must also change the limits of integration:
- When \( x = 0 \), \( u = \sin(0) = 0 \).
- When \( x = \pi/2 \), \( u = \sin(\pi/2) = 1 \).
Substituting these into the integral, we get:
\[ I = \int_0^1 2u e^u du \] This integral can be solved using integration by parts. Let's pull the constant out:
\[ I = 2 \int_0^1 u e^u du \] For the integration by parts, let:
- \( \text{part 1} = u \) (so \( d(\text{part 1}) = du \))
- \( d(\text{part 2}) = e^u du \) (so \( \text{part 2} = \int e^u du = e^u \))
Applying the formula \( \int (\text{part 1}) d(\text{part 2}) = (\text{part 1})(\text{part 2}) - \int (\text{part 2}) d(\text{part 1}) \):
\[ \int u e^u du = u e^u - \int e^u du = u e^u - e^u \] Now we apply this result to our definite integral:
\[ I = 2 [u e^u - e^u]_0^1 \] \[ I = 2 \left( (1 \cdot e^1 - e^1) - (0 \cdot e^0 - e^0) \right) \] \[ I = 2 \left( (e - e) - (0 - 1) \right) \] \[ I = 2 \left( 0 - (-1) \right) \] \[ I = 2(1) = 2 \] Step 4: Final Answer:
The value of the definite integral is 2. This corresponds to option (C).