What is the integral of csc(2θ) ?

The integral of csc(2θ) is 1/2 ln|tan(θ)|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of csc(2θ)

\int csc (2 θ) d θ

\frac{1}{2} ln ∣ tan (θ) ∣ + C

Solution steps

\int csc (2 θ) d θ

Apply u-substitution

: \int csc (u) \frac{1}{2} d u

= \int csc (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int csc (u) d u

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} ln ∣ v ∣

Substitute back

= \frac{1}{2} ln tan (\frac{2 θ}{2})

Simplify

\frac{1}{2} ln tan (\frac{2 θ}{2}) : \frac{1}{2} ln ∣ tan (θ) ∣

= \frac{1}{2} ln ∣ tan (θ) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ tan (θ) ∣ + C

\int x (x + 2) d x

\int \frac{x + 2}{x ^{3} - 1} d x

\int x sin (2 x + 3) d x

\int \frac{x ^{3}}{( 1 + 2 x ^{2} ) ^{3}} d x

\int csc (θ) (csc (θ) + cot (θ)) d θ