Question

${\int }_0^{\pi /2}\frac{\cos e{c}^{2/3}x}{\cos e{c}^{2/3}x+{\sec }^{2/3}x}{d}_{X=}$

• A. $\pi$
• B. $-\pi$
• C. $\pi /4$
• D. $0$

Answer 1

Answer: (C)

$\pi /4$

$I={\int }_0^{\frac{\pi }{2}}\frac{cose{c}^{\frac{2}{3}}}{cose{c}^{\frac{2}{3}}+se{c}^{\frac{2}{3}}}dx$
w. r. t
${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx$
So, $I={\int }_0^{\frac{\pi }{2}}\left(\frac{se{c}^{\frac{2}{3}x}}{cose{c}^{2}3x+se{c}^{2}3x}\right)dx$
$2I={\int }_0^{\frac{\pi }{2}}\left(\frac{se{c}^{\frac{2}{3}}x}{cose{c}^{\frac{2}{3}}x+sse{c}^{\frac{2}{3}}x}\right)dx+{\int }_0^{\frac{\pi }{2}}\frac{cose{c}^{\frac{2}{3}}x}{(cose{c}^{\frac{2}{3}}x+se{c}^{\frac{2}{3}x})}dx$
$2I={\int }_0^{\frac{\pi }{2}}1\cdot dx$

$I=\frac{\pi }{4}$