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

The integral of sqrt((1-cos(x))) is -2sqrt(2)cos(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 sqrt((1-cos(x)))

\int (1 - cos (x)) d x

- 22 cos (\frac{x}{2}) + C

Solution steps

\int 1 - cos (x) d x

Rewrite using trig identities

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

Take the constant out:

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

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

sin^{2} (\frac{x}{2}) = (sin (\frac{x}{2})),

assuming

sin (\frac{x}{2}) \geq 0

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

Apply u-substitution

: \int sin (u) \cdot 2 d u

= 2 \cdot \int sin (u) \cdot 2 d u

Take the constant out:

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

= 2 \cdot 2 \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= 2 \cdot 2 (- cos (u))

Substitute back

u = \frac{x}{2}

= 2 \cdot 2 (- cos (\frac{x}{2}))

Simplify

= - 22 cos (\frac{x}{2})

Add a constant to the solution

= - 22 cos (\frac{x}{2}) + C

\int \frac{1}{x ^{3} - 27} d x

\int \frac{x ^{3} + 7 x ^{2} + 25 x + 35}{x ^{2} + 5 x + 6} d x

\int \frac{4 sinh ( x )}{4 + cosh ^{2} ( x )} d x

\frac{d y}{d x}, y = 3 x

\frac{d}{d x} (x + \frac{9}{x})

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