Before Solve this problem, we have to understand some characteristics of odd functions. Definition :Odd functions A function f is said to be an odd function if -f(x)=f(-x), for all value of x. Observations: The graph of an odd function will be symmetrical about the origin. Example: f(x)=x3 is odd proof: f(-x)=(-x)3=-(x3)=-f(x) Observe that the graph of x3 function is symmetrical about the origin. Now Let's talk about the integrals of odd functions. Theorem Let f be an integrable function on some closed interval that is symmetric about zero. Let's call the interval [-a,a] for a>0. If f is odd function then, ∫a-af(x)dx=0 proof Let f be an integrable function on some closed interval that is symmetric about zero. Let's call the interval [-a,a] for a>0. Suppose that f is odd function. Then, ∫a-af(x)dx=∫0-af(x)dx+∫a0f(x)dx-------(1) Before move on furthe...