Question

If f(x, y) = 0 be the solution of differential equation (2y cosec 2x + ln cot y)dx + (ln tan x - 2x cosec 2y)dy = 0 such that $f(\frac{\pi }{4},\frac{\pi }{2})=0$

If ${\int }_0^{\infty }\frac{(\frac{5\pi }{4},\frac{2017\pi }{4})cosx}{x}dx=\frac{\pi }{2},then{\int }_0^{\infty }\frac{1}{x}(1-si{n}^{2}x{)}^{\frac{3}{2}}dx$ is equal to

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{2}$
• D. $\pi$

Answer 1

The correct option is C

$\frac{\pi }{2}$

${\int }_0^{\infty }\frac{cosx}{x}dx=\frac{\pi }{2}{\int }_0^{\infty }\frac{co{s}^{3}x}{x}dx={\int }_0^{\infty }\frac{3}{4}\frac{cosx}{x}dx+{\int }_0^{\infty }\frac{co{s}^{3}x}{4x}dx=\frac{3}{4}.\frac{\pi }{2}+{\int }_0^{\infty }\frac{cost}{4.\frac{t}{3}}\frac{dt}{3}=\frac{3}{4}.\frac{\pi }{2}+\frac{1}{4}.\frac{\pi }{2}=\frac{\pi }{2}$