We are tasked with evaluating the integral:

$\int_{-1}^{1} \sin(x) \cos^2(x) e^{x^2} \, dx$

Step 1: Analyze the integral

The integrand is $f(x) = \sin(x) \cos^2(x) e^{x^2}$. Since we are integrating from $-1$ to $1$, it's useful to check if the integrand is odd or even. This can sometimes simplify the problem by allowing us to immediately determine if the integral is zero (for odd functions integrated over symmetric limits) or reduce the integration range.

Test for an even or odd function:

A function $f(x)$ is:

• Even if $f(-x) = f(x)$.
• Odd if $f(-x) = -f(x)$.

Let's check $f(-x)$: $f(-x) = \sin(-x) \cos^2(-x) e^{(-x)^2} = -\sin(x) \cos^2(x) e^{x^2}.$ We see that: $f(-x) = -f(x).$ Thus, $f(x) = \sin(x) \cos^2(x) e^{x^2}$ is an odd function.

Step 2: Apply properties of odd functions

Since the function is odd, and we are integrating over a symmetric interval $[-1, 1]$, we can apply the property of odd functions:

$\int_{-a}^{a} f(x) \, dx = 0 \quad \text{if} \quad f(x) \text{ is odd}.$

In this case, the integrand is odd and the interval is symmetric about zero, so we conclude:

$\int_{-1}^{1} \sin(x) \cos^2(x) e^{x^2} \, dx = 0.$

Conclusion:

The value of the integral is $0$, due to the oddness of the function and the symmetric limits of integration.

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Let me know if you'd like further details on how odd and even functions impact integrals or have any other questions!

Related questions:

1. What is the integral of an even function over symmetric limits?
2. How can you simplify integrals with trigonometric functions?
3. How do you handle integrals involving exponential functions?
4. What are some common techniques for testing whether a function is odd or even?
5. What is the physical interpretation of integrals with odd functions?

Tip:

When integrating, always check if the function is odd or even, especially over symmetric intervals. It can save you a lot of work!