Evaluating ∫ sec 2 ⁡ ( x ) ( 4 + tan 2 ⁡ ( x ) ) 2 d x

Question

Evaluating   ∫                    sec        2            ⁡      (      x      )              (      4      +              tan        2            ⁡      (      x      )              )        2                d  x

reinmelk3iu · Accepted Answer

If you want to check your answer, just differentiate and see if you get back the function you started from.
But there's a much simpler method. I'd simply observe that with the substitution

\tan x = 2 u

, the integral becomes

\frac{1}{8} \int \frac{1}{(1 + u^{2})^{2}} d u

which is standard. Leaving aside the factor 1/8, we can write

\int \frac{1 + u^{2} - u^{2}}{(1 + u^{2})^{2}} d u = \int \frac{1}{1 + u^{2}} d u + \frac{1}{2} \int u \frac{- 2 u}{(1 + u)^{2}} d u = \arctan u + \frac{1}{2} \frac{u}{1 + u^{2}} - \frac{1}{2} \int \frac{1}{1 + u^{2}} d u

and therefore

\frac{1}{2} \arctan u + \frac{1}{2} \frac{u}{1 + u^{2}} + c

Now reinsert the factor

1 / 8

and do back substitution to get

\frac{1}{16} (\arctan (\frac{\tan x}{2}) + \frac{2 \tan x}{4 + \tan^{2} x}) + c

bucstar11n0h · Accepted Answer

You can use the noble method known as Ostrogradski method:
Assume you are at the step evaluating

\int \frac{1}{(1 + u^{2})^{2}} d u

. According to this method, it

\int \frac{1}{(1 + u^{2})^{2}} d u = \frac{a u + b}{1 + u^{2}} + \int \frac{c u + d}{1 + u^{2}} d u

Now apply the method to find the constants a,b,c and d.

How to solve this integral? int (sec^2(x))/((4+tan^2(x))^2)dx

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