If f is an integrable function, show that (i) ∫f(x^2)dx, x ∈ [-a, a] = 2 ∫f(x^2)dx, x ∈ [0, a] (ii) ∫xf(x^2)dx, x ∈ [-a, a]

Question

If f is an integrable function, show that

(i) \(\int\limits_{-a}^{a} \)f(x²)dx= 2\(\int\limits_{0}^{a} \)f(x²)dx

(ii)\(\int\limits_{-a}^{a} \)xf(x²)dx = 0

Answer 1

(i) Let us check the given function for being even and odd.

f((–x)²) = f(x²)

The function does not change sign and therefore the function is even.

We know that if f(x) is an even function,

Hence proved.

(ii) Let us check the given function for even and odd.

Let g(x) = xf(x²)

g(–x) = –x f((–x)²)

g(–x) = – xf(x²)

g(–x) = – g(x)

Therefore, the function is odd.

We know that if f(x) is an odd function,

Hence, proved.