What is the integral of 1/(sqrt(1+cos(x))) ?

The integral of 1/(sqrt(1+cos(x))) is sqrt(2)ln|sec(x/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/(sqrt(1+cos(x)))

\int \frac{1}{1 + cos ( x )} d x

2 ln sec (\frac{x}{2}) + tan (\frac{x}{2}) + C

Solution steps

\int \frac{1}{1 + cos ( x )} d x

Apply u-substitution

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

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

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

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

Take the constant out:

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

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

Simplify

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

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

Simplify

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

Use the common integral:

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

= 2 ln u^{2} + 1 + u

Substitute back

u = tan (\frac{x}{2})

= 2 ln tan^{2} (\frac{x}{2}) + 1 + tan (\frac{x}{2})

Simplify

2 ln tan^{2} (\frac{x}{2}) + 1 + tan (\frac{x}{2}) : 2 ln sec (\frac{x}{2}) + tan (\frac{x}{2})

= 2 ln sec (\frac{x}{2}) + tan (\frac{x}{2})

Add a constant to the solution

= 2 ln sec (\frac{x}{2}) + tan (\frac{x}{2}) + C

x \to 0 lim (\frac{1}{x} + \frac{1}{x ^{2}})

n = 1 \sum \infty (2 n)!

n \to 0 lim (ln (n))

\int \frac{x}{x ^{2} + 8} d x

8 x^{\frac{3}{2}} + 6 x^{\frac{1}{2}} + 4^{- \frac{1}{2}}

What is the integral of 1/(sqrt(1+cos(x))) ?
The integral of 1/(sqrt(1+cos(x))) is sqrt(2)ln|sec(x/2)+tan(x/2)|+C