Calculate the indefinite integrals ∫dx/cos^2x(√1+tanx)

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You should solve the indefinite integral using substitution, hence, you should come up with the following substitution such that:

`1 + tan x= u => (dx)/(cos^2 x) = du`

`int (dx)/((cos^2 x)sqrt(1 + tan x)) = int (du)/(sqrt(u)) `

`int (dx)/(cos^2 x)sqrt(1 + tan x) = int u^(-1/2) du`

`int u^(-1/2) du = (u^(-1/2+1))/(-1/2+1) + c`

`int u^(-1/2) du = 2u^(1/2) + c => int u^(-1/2) du = 2sqrt u + c`

Substituting back `1 + tan x`  for u yields:

`int (dx)/((cos^2 x)sqrt(1 + tan x)) = 2sqrt(1 + tan x) + c`

Hence, evaluating the given integral using substitution yields `int (dx)/((cos^2 x)sqrt(1 + tan x)) = 2sqrt(1 + tan x) + c` .

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