Find $\int_{-\pi/2}^{\pi/2} \frac{\cos^3 x}{e^x+1} dx$

Question

Find the value of the following integral

$$ \int_{-\pi/2}^{\pi/2} \frac{\cos^3 x}{e^x+1} dx $$

I tried to use the twin integral with sin and get some result but nothing obvious came out. Also, Mathematica gives a very bizarre result! Any ideas how to solve analytically?

Answer 1

Hint: make the substitution $x\mapsto -x$ and add the two integrals. The result should be $2/3$.

Answer 2

Like Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$.,

if $\displaystyle I=\int_{-a}^a\dfrac{g(x)\ dx}{b^x+1}$

$\displaystyle I=\int_{-a}^a\dfrac{g(a-a-x)\ dx}{b^{a-a-x}+1}$

If $g(x)=g(-x)$ i.e., an even function,

$\displaystyle I=\int_{-a}^ag(x)\ dx$

Here $g(x)=\cos^3x=\dfrac{\cos3x+3\cos x}4$