What is the integral of sqrt(x-x^2) ?

The integral of sqrt(x-x^2) is 1/16 (2arcsin(2x-1)+sin(2arcsin(2x-1)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(x-x^2)

\int x - x^{2} d x

\frac{1}{16} (2 arcsin (2 x - 1) + sin (2 arcsin (2 x - 1))) + C

Solution steps

\int x - x^{2} d x

Complete the square

x - x^{2} : - (x - \frac{1}{2})^{2} + \frac{1}{4}

= \int - (x - \frac{1}{2})^{2} + \frac{1}{4} d x

Apply u-substitution

: \int \frac{- 4 u ^{2} + 1}{2} d u

= \int \frac{- 4 u ^{2} + 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 - 4 u^{2} + 1 d u

Apply Trigonometric Substitution

: \int \frac{1}{2} cos^{2} (v) d v

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

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 cos^{2} (v) d v

Rewrite using trig identities

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

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 \frac{1}{2} \cdot \int 1 + cos (2 v) d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} (arcsin (2 (x - \frac{1}{2})) + \frac{1}{2} sin (2 arcsin (2 (x - \frac{1}{2}))))

Simplify

\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} (arcsin (2 (x - \frac{1}{2})) + \frac{1}{2} sin (2 arcsin (2 (x - \frac{1}{2})))) : \frac{1}{16} (2 arcsin (2 x - 1) + sin (2 arcsin (2 x - 1)))

= \frac{1}{16} (2 arcsin (2 x - 1) + sin (2 arcsin (2 x - 1)))

Add a constant to the solution

= \frac{1}{16} (2 arcsin (2 x - 1) + sin (2 arcsin (2 x - 1))) + C

81 y^{^{''}} - 18 y^{^{'}} + y = 0

\int x^{- 2} sin (\frac{1}{x}) d x

\frac{d P}{d t} = P t, P (1) = 2

\frac{d ^{2}}{d x ^{2}} ((- 5 x + 6)^{5})

t \cdot y^{^{'}} + y = 2 t

What is the integral of sqrt(x-x^2) ?
The integral of sqrt(x-x^2) is 1/16 (2arcsin(2x-1)+sin(2arcsin(2x-1)))+C