What is the integral of cos(2x)cot(2x) ?

The integral of cos(2x)cot(2x) is 1/2 ln|tan(x)|+1/2 cos(2x)+C

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integral of cos(2x)cot(2x)

\int cos (2 x) cot (2 x) d x

\frac{1}{2} ln ∣ tan (x) ∣ + \frac{1}{2} cos (2 x) + C

Solution steps

\int cos (2 x) cot (2 x) d x

Rewrite using trig identities

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

Expand

\frac{1 - sin ^{2} ( 2 x )}{sin ( 2 x )} : \frac{1}{sin ( 2 x )} - sin (2 x)

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

\int \frac{1}{sin ( 2 x )} d x = \frac{1}{2} ln ∣ tan (x) ∣

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

= \frac{1}{2} ln ∣ tan (x) ∣ - (- \frac{1}{2} cos (2 x))

Simplify

= \frac{1}{2} ln ∣ tan (x) ∣ + \frac{1}{2} cos (2 x)

Add a constant to the solution

= \frac{1}{2} ln ∣ tan (x) ∣ + \frac{1}{2} cos (2 x) + C

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What is the integral of cos(2x)cot(2x) ?
The integral of cos(2x)cot(2x) is 1/2 ln|tan(x)|+1/2 cos(2x)+C