Question

Assertion :The value of ${\int }_0^{2\pi }{\cos }^{99}xdx$ is 0. Reason: ${\int }_0^{2a}f\left(x\right)dx=2{\int }_0^{a}f\left(x\right)dx,$ if $f(2a-x)=f\left(x\right)$

• 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. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$\mathrm{Let}I={\int }_0^{2\pi }{\cos }^{99}xdx$
Then, $I=2{\int }_0^{\pi }{\cos }^{99}xdx[\because {\cos }^{99}(2\pi -x)={\cos }^{99}x]$
Now, ${\int }_0^{\pi }{\cos }^{99}xdx=0[\because {\cos }^{99}(\pi -x)=-{\cos }^{99}x]$
$\Rightarrow I=2\times 0=0$