How to use 3 tan 2 ⁡ x = − 3 + 3 sec 2 ⁡ x to integrate 3 tan 2 ⁡ x?

Question

How to use   3      tan    2    ⁡  x  =  −  3  +  3      sec    2    ⁡  x to integrate   3      tan    2    ⁡  x?

Elianna Douglas · Accepted Answer

the pythagorean identity   1  +      tan    2    ⁡  x  =      sec    2    ⁡  x. Then clearly also             tan      2        ⁡    x    =          sec      2        ⁡    x    −    1  , so:  3            tan      2        ⁡    x    =  3      (                  sec        2            ⁡      x      −      1        )    =  3      sec    2    ⁡  x  −  3  =  −  3  +  3      sec    2    ⁡  xThen by linearity of the integral:  ∫  3      tan    2    ⁡  x        d    x  =  ∫  3      sec    2    ⁡  x  −  3        d    x  =  3  ∫      sec    2    ⁡  x        d    x  −  ∫  3        d    x  =  ⋯

deceptie3j · Accepted Answer

We can evaluate the integral using your identity. Basically,

\int 3 \tan^{2} x d x = \int 3 (\sec^{2} x - 1) d x = \int (3 \sec^{2} x - 3) d x = 3 \tan x - 3 x + C

From the fact that

\frac{d}{d x} \tan x = \sec^{2} x ⟺ \int \sec^{2} x d x = \tan x + C

How to use 3 tan 2 </msup> ⁡ x = −<!-- − --

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