Evaluate the Integral

Problem

(∫_−10^10)(sin(x*e^(x^2))*d(x))

Solution

1. Identify the function to be integrated as ƒ(x)=sin(x*e^(x^2))

2. Determine the symmetry of the function by substituting −x for x

3. Evaluate the function at −x ƒ*(−x)=sin((−x)*e^((−x)^2))=sin(−x*e^(x^2))

4. Apply the property of the sine function, which is an odd function, meaning sin(−θ)=−sin(θ)

5. Conclude that ƒ*(−x)=−sin(x*e^(x^2))=−ƒ(x) which proves that ƒ(x) is an odd function.

6. Apply the integral property for odd functions over a symmetric interval [−a,a] which states that (∫_−a^a)(ƒ(x)*d(x))=0

Final Answer

(∫_−10^10)(sin(x*e^(x^2))*d(x))=0

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