${\int }_0^{\mathrm{\pi }/2}\frac{{\cos }^3\mathrm{x}}{\sin \mathrm{x}+\cos \mathrm{x}}\mathrm{dx}=$

$\text{I\hspace{0.17em}=\hspace{0.17em}}\underset{\text{0}}{\overset{\text{π/2}}{\int }}\frac{{\text{cos}}^{\text{3}}\text{x}}{\text{sin\hspace{0.17em}x\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}cosx}}\text{\hspace{0.17em}dx\_\_\_\_\_\_\_}\left(\text{1}\right)$

use $\underset{\text{0}}{\overset{\text{a}}{\int }}\text{f}\left(\text{x}\right)\text{\hspace{0.17em}dx=\hspace{0.17em}\hspace{0.17em}}\underset{\text{0}}{\overset{\text{a}}{\int }}\text{f}\left(\text{a-x}\right)\text{\hspace{0.17em}\hspace{0.17em}dx}$

$\text{I\hspace{0.17em}=\hspace{0.17em}}\underset{\text{0}}{\overset{\text{π/2}}{\int }}\frac{{\text{sin}}^{\text{3}}\text{x}}{\text{cosx\hspace{0.17em}+\hspace{0.17em}sin\hspace{0.17em}x}}\text{dx\_\_\_\_\_\_\_}\left(\text{2}\right)$

Add (1) and (2)

$\text{2I\hspace{0.17em}=\hspace{0.17em}}\underset{\text{0}}{\overset{\text{π/2}}{\int }}\frac{{\text{cos}}^{\text{3}}{\text{x\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}sin}}^{\text{3}}\text{x}}{\text{sinx\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}cosx}}\text{dx}$

$\text{2I\hspace{0.17em}=\hspace{0.17em}}\underset{\text{0}}{\overset{\text{π/2}}{\int }}\left(\text{1-sinx\hspace{0.17em}\hspace{0.17em}cosx}\right)\text{\hspace{0.17em}\hspace{0.17em}dx}$

$\text{2I\hspace{0.17em}= }{\left(\text{x\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}}\frac{\text{1}}{\text{4}}\text{\hspace{0.17em}\hspace{0.17em}cos\hspace{0.17em}\hspace{0.17em}2x}\right)}_{\text{0}}^{\frac{\text{π}}{\text{2}}}$

$\text{2I\hspace{0.17em}= }\left(\frac{\text{π}}{\text{2}}\text{-}\frac{\text{1}}{\text{4}}\right)\text{-}\left(\text{0+}\frac{\text{1}}{\text{4}}\right)$

$\text{2I\hspace{0.17em}= }\frac{\text{π}}{\text{2}}\text{-\hspace{0.17em}}\frac{\text{1}}{\text{2}}$

$\text{I\hspace{0.17em}= }\frac{\text{π-1}}{\text{4}}$