Question

Assertion :Let f is odd function, then value of ${\int }_{-a}^a\frac{f\left(\sin x\right)}{f\left(\cos x\right)+f\left({\sin }^{2}x\right)}dx$ is zero Reason: For odd function ${\int }_{-a}^{a}f\left(x\right)dx=0$

• 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Since the denominator is even & the numerator $f\left(\sin (-x)\right)=f(-\sin x)=-f\left(\sin x\right)$ as $f$ is odd.
$\therefore$ $\frac{f\left(\sin x\right)}{f\left(\cos x\right)+f\left({\sin }^{2}x\right)}$ is an odd function
$\therefore$ ${\int }_{-a}^a\frac{f\left(\sin x\right)}{f\left(\cos x\right)+f\left({\sin }^{2}x\right)}dx=0$