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1-x2 / (1+x2)2 where x= tan θ
(1-tan^2Φ)/(sec^2Φ)^2 now put u=sec^2phi
∫1-x2/(1+x2)2
add and subtract 1 to numerator ,you get
∫2/(1+x2)2 dx - ∫1/(1+x2) dx ===> ∫2/(1+x2)2 dx-tanx+c
= let x=tanθ
===> dx=sec2θ dθ
∫2/(1+tan2θ)2 sec2θ dθ -tanx+c
=∫2cos2θ dθ-tanx+c
=∫(cos2θ+1)dθ-tanx+c
=sin2θ/2 +x-tanx+c
=tanθ/1+tan2θ +x-tanx+c
but x=tanθ
x/(1+x2) +x-tanx+c
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