Question

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

• A. $I_1=I_2$
• B. $3I_1=I_2$
• C. $I_1=3I_2$
• D. $I_1=5I_2$

Answer 1

The correct option is C

$I_1=3I_2$

Explanation for correct option:

Find the relation:

Given that,

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

$I_2={\int }_0^{\mathrm{\pi }}f\left({\cos }^{2}x\right)dx$

Now,

$$\begin{gathered} I_1={\int }_0^{3\mathrm{\pi }}f(\cos ²x)dx \\ =3{\int }_0^{\mathrm{\pi }}f(\cos ²x)dx\left[\because {\int }_0^{nT}f\left(x\right)dx=n{\int }_0^{T}f\left(x\right)dx,\text{f(x) is periodic with}T\right] \\ =3I_2 \end{gathered}$$

Hence, the correct option is C.