What is the integral of (cos^2(2x))/(sin(2x)) ?

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

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integral of (cos^2(2x))/(sin(2x))

\int \frac{cos ^{2} ( 2 x )}{sin ( 2 x )} d x

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

Solution steps

\int \frac{cos ^{2} ( 2 x )}{sin ( 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

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

\int x 2 - 3 x^{2} d x

\int (tan (x) + 1) d x

\int 2 x^{- \frac{1}{7}} + 8 d x

\int \frac{4}{sec ( x ) - 1} d x

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