Question

Assertion :${\int }_0^{2\pi }{\sin }^{3}xdx=0$ Reason: If $f\left(x\right)={\sin }^{3}x$ then $f(-x)=-f\left(x\right)$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$f\left(x\right)={\sin }^{3}x$
$\therefore$ $f(-x)={\left(\sin (-x)\right)}^3=-{\sin }^{3}x=-f\left(x\right)$
$\therefore$ Reason (R) is true
Again Let $I={\int }_0^{2\pi }{\sin }^{3}xdx$
$={\int }_0^{2\pi }{\left(\sin (2\pi -x)\right)}^3$ $(\because {\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x))dx$
$={\int }_0^{2\pi }-{\sin }^{3}xdx$
$2I=0$, $I=0$
$\therefore$ Assertion (A) is also true
But the reason is not the correct explanation for the assertion.