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

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

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

integral

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

\frac{1}{2} ln ∣ sin (x) ∣ + C

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply u-substitution

: \int \frac{1}{u} d u

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = sin (x)

= \frac{1}{2} ln ∣ sin (x) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ sin (x) ∣ + C

(cos (x))^{4}

\frac{x}{( x ^{2} x ^{2} + 1 )}

x^{2} \cdot 4 - x

\frac{e ^{2}}{x ^{2}}

ln (2 + x)

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