What is the integral of sqrt((4-x)x) ?

The integral of sqrt((4-x)x) is -4arcsin(1/2 sqrt(4-x))+sin(4arcsin(1/2 sqrt(4-x)))+C

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integral of sqrt((4-x)x)

\int (4 - x) x d x

- 4 arcsin (\frac{1}{2} 4 - x) + sin (4 arcsin (\frac{1}{2} 4 - x)) + C

Solution steps

\int (4 - x) x d x

Apply exponent rule:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

(4 - x) x = (4 - x) x

= \int 4 - x x d x

Apply Integration By Parts

: \frac{2}{3} x^{\frac{3}{2}} 4 - x - \int - \frac{x ^{\frac{3}{2}}}{3 4 - x} d x

= \frac{2}{3} x^{\frac{3}{2}} 4 - x - \int - \frac{x ^{\frac{3}{2}}}{3 4 - x} d x

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

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

Simplify

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

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

Add a constant to the solution

= - 4 arcsin (\frac{1}{2} 4 - x) + sin (4 arcsin (\frac{1}{2} 4 - x)) + C

y^{^{'}} + 3 y = t + e^{(- 2 t)}

x \to - 3 lim (\frac{1}{( x + 3 ) ^{2}})

y^{^{'}} - e^{x} e^{y} = - 1

\frac{\partial}{\partial x} (e^{1 + x^{2} - y^{2}})

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

What is the integral of sqrt((4-x)x) ?
The integral of sqrt((4-x)x) is -4arcsin(1/2 sqrt(4-x))+sin(4arcsin(1/2 sqrt(4-x)))+C