What is the integral of (cot^2(2x))/(sec(2x)) ?

The integral of (cot^2(2x))/(sec(2x)) is -1/2 sin(2x)+1/4 (-cot(x)-tan(x))+C

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integral of (cot^2(2x))/(sec(2x))

\int \frac{cot ^{2} ( 2 x )}{sec ( 2 x )} d x

- \frac{1}{2} sin (2 x) + \frac{1}{4} (- cot (x) - tan (x)) + C

Solution steps

\int \frac{cot ^{2} ( 2 x )}{sec ( 2 x )} d x

Rewrite using trig identities

= \int \frac{- 1 + csc ^{2} ( 2 x )}{sec ( 2 x )} d x

Expand

\frac{- 1 + csc ^{2} ( 2 x )}{sec ( 2 x )} : - \frac{1}{sec ( 2 x )} + \frac{csc ^{2} ( 2 x )}{sec ( 2 x )}

= \int - \frac{1}{sec ( 2 x )} + \frac{csc ^{2} ( 2 x )}{sec ( 2 x )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{1}{sec ( 2 x )} d x + \int \frac{csc ^{2} ( 2 x )}{sec ( 2 x )} d x

\int \frac{1}{sec ( 2 x )} d x = \frac{1}{2} sin (2 x)

\int \frac{csc ^{2} ( 2 x )}{sec ( 2 x )} d x = \frac{1}{4} (- cot (x) - tan (x))

= - \frac{1}{2} sin (2 x) + \frac{1}{4} (- cot (x) - tan (x))

Add a constant to the solution

= - \frac{1}{2} sin (2 x) + \frac{1}{4} (- cot (x) - tan (x)) + C

\int \frac{4 x ^{2}}{( x - 2 ) ^{3}} d x

\int \frac{sin ( \frac{1}{x ^{6}} )}{x ^{7}} d x

\frac{d}{d x} (\frac{1}{2} cos^{2} (x))

f (x) = \frac{3}{x ^{\frac{1}{2}}}

x \to 3 + lim (\frac{x + 2}{x + 3})

What is the integral of (cot^2(2x))/(sec(2x)) ?
The integral of (cot^2(2x))/(sec(2x)) is -1/2 sin(2x)+1/4 (-cot(x)-tan(x))+C