Question

Assertion :${\int }_a^{x}f\left(t\right)dt$ is an even function if $f\left(x\right)$ is an odd function. Reason: ${\int }_a^{x}f\left(t\right)dt$ is an odd function if $f\left(x\right)$ is an even function.

• A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1.
• B. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1.
• C. STATEMENT-1 is True, STATEMENT-2 is False.
• D. STATEMENT-1 is False, STATEMENT-2 is True.

Answer 1

The correct option is C STATEMENT-1 is True, STATEMENT-2 is False.

Statement 1 is true as it is a fundamental property. (See integration of odd and even functions.)

Let $g\left(x\right)={\int }_a^{x}f\left(t\right)dt$

If $f\left(x\right)$ is an even function, then

$g(-x)={\int }_a^{-x}f\left(t\right)dt$

$=-{\int }_{-a}^{x}f(-y)dy$

$=-{\int }_{-a}^{x}f\left(y\right)dy$

$=-{\int }_{-a}^{a}f\left(y\right)dy-{\int }_a^{x}f\left(y\right)dy$

$\ne -g\left(x\right)$

Hence, statement $2$ is false.