 # It's on Indefinite Integrals int sqrt(1+2 tan x (tan x + sec x )) dx Please tell me the way of solving such questions. like what could i assume sec x or sec x tan x to be equal to? Wevybrearttexcl 2022-09-17 Answered
It's on Indefinite Integrals
$\int \sqrt{1+2\mathrm{tan}x\left(\mathrm{tan}x+\mathrm{sec}x\right)}dx$
Please tell me the way of solving such questions. like what could i assume sec x or sec x tan x to be equal to?
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$\mathbf{M}\mathbf{y}\phantom{\rule{thickmathspace}{0ex}}\mathbf{S}\mathbf{o}\mathbf{l}\mathbf{u}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{::}$ Let
$I=\int \sqrt{1+2\mathrm{tan}x\left(\mathrm{tan}x+\mathrm{sec}x\right)}dx=\int \sqrt{1+2{\mathrm{tan}}^{2}\left(x\right)+2\mathrm{tan}x\cdot \mathrm{sec}x}dx$
So $I=\int \sqrt{1+{\mathrm{tan}}^{2}x+{\mathrm{tan}}^{2}x+2\mathrm{tan}x\cdot \mathrm{sec}x}dx=\int \sqrt{{\left(\mathrm{tan}x+\mathrm{sec}x\right)}^{2}}dx$
So $I=\int \mathrm{tan}xdx+\int \mathrm{sec}xdx=\mathrm{ln}|\mathrm{sec}x|+\mathrm{ln}|\mathrm{sec}x+\mathrm{tan}x|+\mathbb{C}=\mathrm{ln}|\mathrm{sec}x\cdot \left(\mathrm{sec}x+\mathrm{tan}x\right)|+\mathbb{C}$

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