Question

If $I_1={\int }_0^{3\pi }f\left({\cos }^{2}x\right)dx$ and $I_2={\int }_0^{\pi }f\left({\cos }^{2}x\right)dx$ then

• A. $I_1=I_2$
• B. $I_1=3I_2$
• C. $I_2=3I_1$
• D. None of these

Answer 1

The correct option is B $I_1=3I_2$

we have $f(x+n\pi )=f\left(x\right)\forall$ integral value of $n$
then ${\int }_0^{n\pi }f\left(x\right)=n{\int }_0^{\pi }f\left(x\right)dx$
Let $g\left(x\right)=f\left({\cos }^{2}x\right)$
$g(x+n\pi )=f\left({\cos }^2(x+n\pi )\right)=f\left({\cos }^{2}x\right)=g\left(x\right)$
$\therefore {\int }_0^{3\pi }f\left({\cos }^{2}x\right)dx=3{\int }_0^{\pi }f\left({\cos }^{2}x\right)dx\Rightarrow I_1=3I_2$