Evaluate ∫ xtan x/ secx cosec x dx, x∈[π,0]

Question

Evaluate \(\int\limits_0^{\pi} \frac{x\, tan x}{sec x \,cosec x}dx.\)

Evaluate ∫ xtan x/ secx cosec x dx, x∈[π,0]

Answer 1

 Let I = \(\int\limits_0^{\pi} \frac{x\, tan x}{sec x \,cosec x}dx\) = \(\int\limits_0^{\pi} \cfrac{x\, \frac{sinx}{cosx}}{\cfrac{1}{sinx cosx}}dx\)

(\(\because\) tan = \(\frac{sin x}{cos x}, sec x = \frac{1}{cos x} and\, cosec\, x = \frac{1}{sinx}\) )

Hence, = \(\int\limits_0^{\pi} \frac{x\, tan x}{sec x \,cosec x}dx\) = \(\frac{\pi^2}{4}\)