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

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

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

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

- ln ∣ tan (x) ∣ + 2 ln ∣ sin (x) ∣ + C

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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

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

= - ln ∣ tan (x) ∣ + 2 ln ∣ sin (x) ∣

Add a constant to the solution

= - ln ∣ tan (x) ∣ + 2 ln ∣ sin (x) ∣ + C

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