Question

integration tan³x dx​

Answer 1

Answer:

Step-by-step explanation:

$\\\tt \int\limits tan^3x \, dx \\= \int\limits (tan^2x )(tan x) \, dx \\= \int\limits (sec^2x-1 )(tan x) \, dx \\= \int\limits sec^2x (tan x) \, dx - \int\limits tan x \, dx$

Now lets split this into two parts just to make it look clean:

Let

y = sex x

then dy = secx tanx dx

Now solving the first term:

 $\\\tt \int\limits y (sec x \ tanx) \, \dfrac{dy}{(sec x \ tanx) }\\\\=\int\limits y dy \\\\=\dfrac{y^{2} }{2} + C_{1}\\=\dfrac{sec^{2}x }{2} + C_{1}$

For the second part:

Let t = cos x

then dt = sin x dx

Now,

$\\\tt\\ \int\limits tan x \, dx \\ = \int\limits \dfrac{sin x }{cos x} \, dx \\= \int\limits \dfrac{sin x }{t} \, \dfrac{dt }{sin x} \\\\= \int\limits \dfrac{1 }{t} dt \\\\= log_{e} |cos x| + C_{2}$

Combining these two gives

$\\\tt \int\limits tan^3x \, dx = \dfrac{sec^{2}x }{2} - log_{e} |cos x| +C$