Question

The value of ${\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\frac{{\cos }^{2}x}{1+3^x}dx$ is:

• A. $2\mathrm{\pi }$
• B. $4\mathrm{\pi }$
• C. $\frac{\mathrm{\pi }}{2}$
• D. $\frac{\mathrm{\pi }}{4}$

Answer 1

The correct option is D

$\frac{\mathrm{\pi }}{4}$

Solve the given integral:

Step 1: Rewrite the given integral.

In the question, an integral ${\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\frac{{\cos }^{2}x}{1+3^x}dx$ is given.

Assume that, $I={\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\frac{{\cos }^{2}x}{1+3^x}dx...1$.

We know that, ${\int }_a^{b}f\left(x\right)dx={\int }_a^b\left[f(a+b-x)\right]dx$.

So,

$$\begin{gathered} I={\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\frac{{\cos }^2\left(-{\displaystyle \frac{\mathrm{\pi }}{2}}+{\displaystyle \frac{\mathrm{\pi }}{2}}-x\right)}{1+3^{\left(-\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{2}-x\right)}}dx \\ \Rightarrow I={\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\frac{{\cos }^2\left(-x\right)}{1+3^{\left(-x\right)}}dx \\ \Rightarrow I={\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\frac{3^x{\cos }^2\left(x\right)}{1+3^x}dx...2 \end{gathered}$$

Step 2: Find the value of the given integral.

Add equation $1$ and equation $2$.

$$\begin{gathered} 2I={\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\left[\frac{{\cos }^2\left(x\right)}{1+3^x}+\frac{3^x{\cos }^2\left(x\right)}{1+3^x}\right]dx \\ \Rightarrow I=\frac{{\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\left[{\cos }^{2}x\right]dx}{2} \end{gathered}$$

We know that, ${\cos }^{2}x=\frac{1+\cos 2x}{2}$.

Therefore,

$$\begin{gathered} I=\frac{{\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\left[{\displaystyle \frac{1+\cos 2x}{2}}\right]dx}{2} \\ \Rightarrow I={\left[\frac{x+{\displaystyle \frac{\sin 2x}{2}}}{4}\right]}_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}} \\ \Rightarrow I=\left[\frac{\mathrm{\pi }}{8}+\frac{\mathrm{\pi }}{8}\right] \\ \Rightarrow I=\frac{\mathrm{\pi }}{4} \end{gathered}$$

Therefore, the value of the given integral is $\frac{\mathrm{\pi }}{4}$.

Hence, option (D) is the correct answer.