Let

|x|=-x, if x<0

And |x|=x, if x≥0

Therefore f(x)=x|x|=-x², if x<0

And f(x)=x|x|=x², if x≥0

Consider x≥0 ⇒ f(x)=x²

Then -x<0 ⇒ f(-x) = -(-x)² = -f(-x)

Now Consider x<0 ⇒ f(x)=-x²

Then -x≥0 ⇒ f(-x) =-(-x)²=x²=-f(x)

Hence f(x) is an odd function. An odd function is a function which satisfies the property f(-x) =-f(-x), ∀ x∈ Domain of f(x)

There is a property of integration of odd functions which states that

if f(x) is an odd function.

Therefore