Question

Assertion :$f\left(x\right)=\cos x$ and $g\left(x\right)=\int \cos xdx$ are both periodic functions Reason: If a function is periodic, then its antiderivative is also a periodic function

• 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 C Assertion is correct but Reason is incorrect

$f\left(x\right)=\cos x$
$g\left(x\right)=\int \cos xdx$
$\Rightarrow g\left(x\right)=\sin x$
We know that sine and cosine functions both are periodic function having a period of $2\pi$
Hence, f(x) and g(x) are periodic functions
Reason is incorrect.
Its not necessary that if a function is periodic, then its antiderivative is also periodic.
For example the function $1+\cot x$ is periodic. But its antiderivative $x+{\log }_e|\sin x|$ is not periodic.