Question

Evaluate $\underset{0}{\overset{\pi /2}{\int }}\frac{co{s}^{3}x}{co{s}^{3}x+si{n}^{3}x}dx$.

Answer 1

$I={\int }_0^{\left(\frac{\pi }{2}\right)}\left(\frac{co{s}^{3}x}{co{s}^{3}x+si{n}^{3}x}\right)={\int }_0^{\left(\frac{\pi }{2}\right)}\left(\frac{co{s}^3\left(\right(\frac{\pi }{2})-x)}{co{s}^3\left(\right(\frac{\pi }{2})-x)+si{n}^3\left(\right(\frac{\pi }{2})-x)}\right)dx={\int }_0^{\left(\frac{\pi }{2}\right)}\left(\frac{si{n}^{3}x}{si{n}^{3}x+si{n}^{3}x}\right)dx\therefore I+I={\int }_0^{\left(\frac{\pi }{2}\right)}\left(\frac{si{n}^{3}x+co{s}^{3}x}{si{n}^{3}x+co{s}^{3}x}\right)dx\therefore I=\left(\frac{1}{2}\right){\int }_0^{\left(\frac{\pi }{2}\right)}dx=\left(\frac{\pi }{4}\right)$