Question

evaluate the definite integral from -pi/2 to pi/2 of( x^2 sinx)/(1+x^6) dx

Answer 1

First examine the function f(x)=x^2sin(x)/(1+x^6). f(-x)=-x^2sin(x)/(1+x^6), so f(x)=-f(-x). The limits are (pi)/2 and -(pi)/2 and the definite integral is the area under the curve between these limits.  But from the analysis of the function, all its positive values cancel out its negative values, with a net result of zero. Hence the definite integral is zero. There is no need to integrate the function.