Question

evaluate the integration
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Answer 1

I hope it can help you:)))

Answer 2

AnswEr :

Given Expression,

$\displaystyle \sf \int \dfrac{tan \: x}{sec \: x + tan \: x} dx$

Multiplying Dividing by sec x - tan x, we obtain :

$\longrightarrow \displaystyle \sf \int \dfrac{tan \: x(sec \: x - tan \: x)}{(sec \: x + tan \: x)(sec \: x - tan \: x)} dx \\ \\ \longrightarrow \displaystyle \sf \int \dfrac{tan \: x.sec \: x - tan {}^{2} \: x}{sec {}^{2} \: x - tan {}^{2} \: x} dx$

Since, sec²x - tan²x = 1

Therefore,

$\longrightarrow \displaystyle \sf \int( sec \: x.tan \: x + 1 - sec {}^{2} x)dx \\ \\ \longrightarrow\displaystyle \sf \int sec \: x.tan \: x.dx \: + \int \: dx \: - \int {sec}^{2} x.dx \\ \\ \longrightarrow \boxed{ \boxed{\sf \: sec \: x + x - tan \: x + C}}$

Integral of the above integrand is x + sec x - tan x + C