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

The integral of (cos(2x))/(2sin(2x)) is 1/4 ln|sin(2x)|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\frac{1}{4} ln ∣ sin (2 x) ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 u} d u

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

Take the constant out:

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

= \frac{1}{2} \cdot \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} \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = sin (2 x)

= \frac{1}{2} \cdot \frac{1}{2} ln ∣ sin (2 x) ∣

Simplify

\frac{1}{2} \cdot \frac{1}{2} ln ∣ sin (2 x) ∣ : \frac{1}{4} ln ∣ sin (2 x) ∣

= \frac{1}{4} ln ∣ sin (2 x) ∣

Add a constant to the solution

= \frac{1}{4} ln ∣ sin (2 x) ∣ + C

\int (4 x e^{x^{2}}) d x

\int x^{1.3} d x

\int 7 x x^{2} + 5 d x

\int - 15 x^{4} (- 3 x^{5} + 6)^{4} d x

\int \frac{1}{x - 2} \frac{x - 1}{2 - x} d x

What is the integral of (cos(2x))/(2sin(2x)) ?
The integral of (cos(2x))/(2sin(2x)) is 1/4 ln|sin(2x)|+C