e^ 2 x [4] e^x+1 d x=

Step-by-step solution

I= e^ 2 x [4] e^x+1 d x Let (e^x+1 )^ 1 4 =t e^ 2 x = (t^4-1 )^2 Differentiating w.r.t. x, 2 e^ 2 x d x=2 (t^4-1 ) 4 t^3 d t aligned & I = (t^4-1 ) (4 t^3 ) d t t I =4 (t^4-1 ) (t^2 ) d t & I =4 (t^6-t^2 ) d t I =4 [ t^7 7 - t^3 3 ]+ C aligned aligned & I =4 t^3 [ t^4 7 - 1 3 ]+ C & I =4 (e^x+1 )^ 3 4 [ (e^x+1 ) 7 - 1 3 ]+ C & I = 4 21 (e^x+1 )^ 3 4 (3 e^x-4 )+ C aligned