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:
Answer & explanation
Correct answer: option 1
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).