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

The integral of xsqrt(-x^2+x) is 1/96 (-32(x(-x+1))^{3/2}+3(2arcsin(2x-1)+sin(2arcsin(2x-1))))+C

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integral of xsqrt(-x^2+x)

\int x - x^{2} + x d x

\frac{1}{96} (- 32 (x (- x + 1))^{\frac{3}{2}} + 3 (2 arcsin (2 x - 1) + sin (2 arcsin (2 x - 1)))) + C

Solution steps

\int x - x^{2} + x d x

Complete the square

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Expand

(2 u + 1) - 4 u^{2} + 1 : 2 u - 4 u^{2} + 1 + - 4 u^{2} + 1

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

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}{4} (\int 2 u - 4 u^{2} + 1 d u + \int - 4 u^{2} + 1 d u)

\int 2 u - 4 u^{2} + 1 d u = - \frac{1}{6} (- 4 u^{2} + 1)^{\frac{3}{2}}

\int - 4 u^{2} + 1 d u = \frac{1}{4} (arcsin (2 u) + \frac{1}{2} sin (2 arcsin (2 u)))

= \frac{1}{4} (- \frac{1}{6} (- 4 u^{2} + 1)^{\frac{3}{2}} + \frac{1}{4} (arcsin (2 u) + \frac{1}{2} sin (2 arcsin (2 u))))

Substitute back

u = x - \frac{1}{2}

= \frac{1}{4} - \frac{1}{6} (- 4 (x - \frac{1}{2})^{2} + 1)^{\frac{3}{2}} + \frac{1}{4} (arcsin (2 (x - \frac{1}{2})) + \frac{1}{2} sin (2 arcsin (2 (x - \frac{1}{2}))))

Simplify

\frac{1}{4} - \frac{1}{6} (- 4 (x - \frac{1}{2})^{2} + 1)^{\frac{3}{2}} + \frac{1}{4} (arcsin (2 (x - \frac{1}{2})) + \frac{1}{2} sin (2 arcsin (2 (x - \frac{1}{2})))) : \frac{1}{96} (- 32 (x (- x + 1))^{\frac{3}{2}} + 3 (2 arcsin (2 x - 1) + sin (2 arcsin (2 x - 1))))

= \frac{1}{96} (- 32 (x (- x + 1))^{\frac{3}{2}} + 3 (2 arcsin (2 x - 1) + sin (2 arcsin (2 x - 1))))

Add a constant to the solution

= \frac{1}{96} (- 32 (x (- x + 1))^{\frac{3}{2}} + 3 (2 arcsin (2 x - 1) + sin (2 arcsin (2 x - 1)))) + C

\frac{d y}{d x} = \frac{y ^{2}}{e ^{x}}, y (0) = - 2

y = x^{2}, (0, - 4)

x \to \infty lim (\frac{x}{ln ( x )})

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

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

What is the integral of xsqrt(-x^2+x) ?
The integral of xsqrt(-x^2+x) is 1/96 (-32(x(-x+1))^{3/2}+3(2arcsin(2x-1)+sin(2arcsin(2x-1))))+C