Question

sec
2
(x)cos
2
(2x) its intregation​

Answer 1

Step-by-step explanation:

We have,

$\int \sec^{2} (x) \cos^{2} (2x)dx$

$= \int \sec^{2} (x) (2 \cos^{2} (x) - 1)^{2} dx$

$= \int \frac{ (2 \cos^{2} (x) - 1)^{2}}{ \cos^{2} (x) } dx$

$= \int \frac{ 4\cos^{4} (x) + 1 - 4 \cos^{2}(x)}{ \cos^{2} (x) } dx$

$= \int ( 4\cos^{2} (x) + \sec^{2} (x) - 4) dx$

$= \int 4\cos^{2} (x) dx + \int \sec^{2} (x)dx - \int4 dx$

$=2 \int 2\cos^{2} (x) dx + \int \sec^{2} (x)dx - \int4 dx$

$=2 \int (1 + \cos(2x) )dx + \int \sec^{2} (x)dx - \int4 dx$

$=2 \int dx +2 \int\cos(2x) dx + \int \sec^{2} (x)dx - \int4 dx$

$=2x + \sin(2x) + \tan (x) - 4x + C$

$\sin(2x) + \tan (x) - 2x + C$