Mathematics - Definite Integration Question with Solution | TestHub

Statement-1: $f(x)=|x|\mathrm{sgn}x$. Since $|x|$ is even and $\mathrm{sgn}x$ is odd, $f(x)$ is an odd function.

For an odd function, ${\int }_{-a}^{a}f(x)dx=0$. Thus, Statement-1 is TRUE.

Statement-2: For a continuous odd function $f(x)$, ${\int }_{-a}^{a}f(x)dx=0$. This is a fundamental property of definite integrals. Thus, Statement-2 is TRUE.

Statement-2 correctly explains Statement-1 because the integrand in Statement-1 is an odd function over a symmetric interval.

The final answer is A