How to tell what is value of integral I = integral (-1 to 1) x^2013e^(-x^2)dx without calculate integral?

Expert Answers
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You may use the following properties of functions and definite integrals, such that:

`int_(-a)^a f(x)dx = 0` if f(x) is odd function

`int_(-a)^a f(x)dx = 2int_0^a f(x)dx` if f(x) even function

Hence, you need to test if the function is odd or even, or a combination of both.

You need to use the definitions of odd and even functions, such that:

`- if f(x) = f(-x) => f(x)` is even

`- if f(-x) = -f(x) => f(x)` is odd

Reasoning by analogy, you need to replace -x for x in equation of the function, such that:

`f(-x) = (-x)^(2013)*e^(-(-x)^2) => f(-x) = -x^2013*e^(-x^2)`

Since `f(x) = x^2013*e^(-x^2)` yields that `f(-x) = -f(x)` , hence the given function is odd.

Since the function is odd, then you do not need to evaluate the definite integral, anymore because, by definition, `int_(-1)^1 x^2013*e^(-x^2)dx = 0` .