Question Number 103860 by mathmax by abdo last updated on 17/Jul/20

find ∫ (dx/(cos^4 x))

Answered by Dwaipayan Shikari last updated on 17/Jul/20

∫(dx/(cos^4 x))=∫sec^4 xdx=∫sec^2 x(1+tan^2 x)dx=∫(1+t^2 )dt=t+(t^3 /3)+C  tanx+((tan^3 x)/3)+C           {put t=tanx

Answered by mathmax by abdo last updated on 17/Jul/20

I =∫  (dx/(cos^4 x))  we know  1+tan^2 x =(1/(cos^2 x)) ⇒(1/(cos^4 x)) =(1+tan^2 x)^2 ⇒  I =∫   (1+tan^2 x)^2  dx =_(tanx =u)    ∫ (1+u^2 )^2 (du/(1+u^2 )) =∫ (1+u^2 )du  =u +(u^3 /3)  +C =tanx +(1/3)tan^3 x +C .