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)=x^3` is odd proof: `f(-x)=(-x)^3=-(x^3)=-f(x)` Observe that the graph of `x^3` 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, `\int_{-a}^{a}f(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, `\int_{-a}^{a}f(x)dx =\int_{-a}^{0}f(x)dx + \int_{0}^{a}f(x)dx-------(1)` Before move on further, Claim: `\int_{x=-a}^{0}f(x)dx = -