The value of which of the following integrals is zero?

(A) $\int\limits_0^1x\, dx$
(B) $\int\limits_{-1}^1x\, dx$
(C) $\int\limits_{-1}^1x^2\, dx$
(D) $\int\limits_0^1\log(\frac{x}{1-x})\, dx$

Choose the correct answer from the options given below:

(B) and (D) only

The correct answer is Option (1) → (B) and (D) only

(A) $\int_{0}^{1}x\,dx$

$=\frac{1}{2}$

Not zero

(B) $\int_{-1}^{1}x\,dx$

$x$ is an odd function and limits are symmetric

Value $=0$

(C) $\int_{-1}^{1}x^2\,dx$

$x^2$ is even, value is positive

Not zero

(D) $\int_{0}^{1}\log\left(\frac{x}{1-x}\right)dx$

Use property $\int_{0}^{1}f(x)dx=\int_{0}^{1}f(1-x)dx$

$\log\left(\frac{1-x}{x}\right)=-\log\left(\frac{x}{1-x}\right)$

Hence integral is zero

The integrals with zero value are (B) and (D).