The discussion revolves around evaluating the integral of sin^7 x/(1+x^10) from -π/2 to π/2. Participants are exploring the properties of the integrand and its behavior over the symmetric interval.
The discussion has progressed towards understanding the characteristics of the integrand, with several participants affirming that sin^7 x is an odd function. The implications of this property on the integral's value are being explored, with some participants suggesting that the product of the functions involved leads to an odd function overall.
Participants are considering the symmetry of the integral's limits and the properties of odd and even functions as part of their reasoning process. There is a recognition of the need for a clear explanation of why integrating an odd function over a symmetric interval results in zero.
Math9999 said:Homework Statement
Find the integral of sin^7 x/(1+x^10) dx from -pi/2 to pi/2.
Homework Equations
None.
The Attempt at a Solution
sin^7 x means sinx to the 7th power. But how do I find this strange integral? I don't think u-substitution, trig identity, any of them will work.
Do you know what even and odd functions are?Math9999 said:I don't know anything about the integrand.
Math9999 said:I know that the sine functions are odd, right?
Math9999 said:That sin^7 (x) is also odd.
Math9999 said:An even function?
Exactly. And what do you get when you integrate an odd function from -a to a?Math9999 said:An odd function.
Math9999 said:0?
Math9999 said:0?