Question

integrate root of tanx with respect to dx in simple steps

Answer 1

(tan x) dx

Let tan x = t2

⇒ sec2 x dx = 2t dt

⇒ dx = [2t / (1 + t4)]dt

⇒ Integral  ∫ 2t2 / (1 + t4) dt

⇒ ∫[(t2 + 1) + (t2 - 1)] / (1 + t4) dt

⇒ ∫(t2 + 1) / (1 + t4) dt + ∫(t2 - 1) / (1 + t4) dt

⇒ ∫(1 + 1/t2 ) / (t2 + 1/t2 ) dt + ∫(1 - 1/t2 ) / (t2+ 1/t2 ) dt

⇒ ∫(1 + 1/t2 )dt / [(t - 1/t)2 + 2] + ∫(1 - 1/t2)dt / [(t + 1/t)2 -2]

Let t - 1/t = u for the first integral ⇒ (1 + 1/t2)dt = du

and t + 1/t = v for the 2nd integral ⇒ (1 - 1/t2)dt = sb