Mathematics

If $$\displaystyle f(x)=p(x)q(x)$$ where $$p(x)=\sqrt{cos\:x},g(x)=log\left(\frac{(1-x)}{1+x}\right)$$ the $$\displaystyle \int_{a}^{b}\:f(x)\:dx$$ equals where $$a=-1/2,b=1/2$$


ANSWER

$$0$$


SOLUTION
$$p(x)=\sqrt{cos\:x}$$
$$p(-x)=\sqrt{cos\:x}=p(x)\therefore p(x)$$ is an even function 
$$\displaystyle q(x)=log\left(\frac{1+x}{1-x}\right)$$
$$\displaystyle q(-x)=log\left(\frac{1-x}{1+x}\right)=-q(x)\therefore q(x)$$ is an odd function 
Now $$f(x)=p(x)q(x)=$$ Product of even and odd function 
                           =an odd function 
$$\displaystyle \therefore \int_{a}^{b}f(x)f(x)dx$$  is equal 0, as a,b  are opposite in sign equal in magnitude
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Single Correct Medium Published on 17th 09, 2020
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