What is the integral of 1/(3-cos(x)) ?

The integral of 1/(3-cos(x)) is 1/(sqrt(2))arctan(sqrt(2)tan(x/2))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(3-cos(x))

\int \frac{1}{3 - cos ( x )} d x

\frac{1}{2} arctan (2 tan (\frac{x}{2})) + C

Solution steps

\int \frac{1}{3 - cos ( x )} d x

Apply u-substitution

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

= \int \frac{1}{2 u ^{2} + 1} d u

Apply Integral Substitution

= \int \frac{1}{2 ( v ^{2} + 1 )} d v

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{1}{2} arctan (v)

Substitute back

= \frac{1}{2} arctan (2 tan (\frac{x}{2}))

Add a constant to the solution

= \frac{1}{2} arctan (2 tan (\frac{x}{2})) + C

\int 3 (8 y - 1) e^{4 y^{2} - y} d y

\int (- t) d t

\int \frac{( e ^{x} )}{( 6 - e ^{x} ) ^{2}} d x

\int sin^{5} (\frac{x}{2}) cos (\frac{x}{2}) d x

What is the integral of 1/(3-cos(x)) ?
The integral of 1/(3-cos(x)) is 1/(sqrt(2))arctan(sqrt(2)tan(x/2))+C