Question

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

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{4}$
• C. $\pi$
• D. $0$

Answer 1

Answer: (B)

$\frac{\pi }{4}$

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

${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx$

$I={\int }_0^{\frac{\pi }{2}}\frac{{\sin }^{\frac{3}{2}}x}{{\sin }^{\frac{3}{2}}x+{\cos }^{\frac{3}{2}}x}={\int }_0^{\frac{\pi }{2}}\frac{{\cos }^{\frac{3}{2}}x}{{\sin }^{\frac{3}{2}}x+co{s}^{\frac{3}{2}}x}=I$

$2I={\int }_0^{\frac{\pi }{2}}1\cdot dx$

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

${\int }_0^{\frac{\pi }{2}}\frac{{\sin }^{\frac{3}{2}}x}{{\sin }^{\frac{3}{2}}x+{\cos }^{\frac{3}{2}}x}dx=\frac{\pi }{4}$