0-pi-2-x-sec-x-csc-x-dx-

Question Number 130958 by bramlexs22 last updated on 31/Jan/21

\int_{0}^{π / 2} \frac{x}{\sec x + \csc x} d x

Commented by benjo_mathlover last updated on 31/Jan/21

M = \underset{0}{\int^{π / 2}} \frac{x}{\sec x + \csc x} dx = \underset{0}{\int^{π / 2}} \frac{\frac{π}{2} - x}{\sec x + \csc x} dx

2 M = \frac{π}{2} \underset{0}{\int^{π / 2}} \frac{dx}{\sec x + \csc x}; M = \frac{π}{4} {\int^{π / 2}}_{0} \frac{\sin x . \cos x}{\sin x + \cos x} dx

M = \frac{π}{4} \int {\overset{π / 2}{}}_{0} \frac{\sin x . \cos x}{\sqrt{2} \sin (x + \frac{π}{4})} dx

Commented by benjo_mathlover last updated on 31/Jan/21

M = \frac{π}{4 \sqrt{2}} \underset{π / 4}{\int^{3 π / 4}} \frac{\frac{1}{\sqrt{2}} (\sin t - \cos t) . \frac{1}{\sqrt{2}} (\cos t + \sin t)}{\sin t} dt

M = \frac{π}{8 \sqrt{2}} \underset{π / 4}{\int^{3 π / 4}} \frac{{2 \sin}^{2} t - 1}{\sin t} dt = \frac{π}{8 \sqrt{2}} {(- 2 \cos t + \ln ∣ \csc t + \cot t ∣)}_{π / 4}^{3 π / 4}

M = \frac{π}{4 \sqrt{2}} [\sqrt{2} + \ln (\sqrt{2} - 1)]