What is the integral of sqrt(x(6-x)) ?

The integral of sqrt(x(6-x)) is -9arcsin(sqrt(1/6 (-x+6)))+9/4 sin(4arcsin(sqrt(1/6 (-x+6))))+C

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integral of sqrt(x(6-x))

\int x (6 - x) d x

- 9 arcsin (\frac{1}{6} (- x + 6)) + \frac{9}{4} sin (4 arcsin (\frac{1}{6} (- x + 6))) + C

Solution steps

\int x (6 - x) d x

Apply exponent rule:

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

x (6 - x) = x (6 - x)

= \int x 6 - x d x

Apply Integration By Parts

: \frac{2}{3} x^{\frac{3}{2}} 6 - x - \int - \frac{x ^{\frac{3}{2}}}{3 6 - x} d x

= \frac{2}{3} x^{\frac{3}{2}} 6 - x - \int - \frac{x ^{\frac{3}{2}}}{3 6 - x} d x

\int - \frac{x ^{\frac{3}{2}}}{3 6 - x} d x = \frac{1}{12} (8 x^{\frac{3}{2}} - x + 6 + 27 (4 arcsin (\frac{- x + 6}{6}) - sin (4 arcsin (\frac{- x + 6}{6}))))

= \frac{2}{3} x^{\frac{3}{2}} 6 - x - \frac{1}{12} (8 x^{\frac{3}{2}} - x + 6 + 27 (4 arcsin (\frac{- x + 6}{6}) - sin (4 arcsin (\frac{- x + 6}{6}))))

Simplify

\frac{2}{3} x^{\frac{3}{2}} 6 - x - \frac{1}{12} (8 x^{\frac{3}{2}} - x + 6 + 27 (4 arcsin (\frac{- x + 6}{6}) - sin (4 arcsin (\frac{- x + 6}{6})))) : - 9 arcsin (\frac{1}{6} (- x + 6)) + \frac{9}{4} sin (4 arcsin (\frac{1}{6} (- x + 6)))

= - 9 arcsin (\frac{1}{6} (- x + 6)) + \frac{9}{4} sin (4 arcsin (\frac{1}{6} (- x + 6)))

Add a constant to the solution

= - 9 arcsin (\frac{1}{6} (- x + 6)) + \frac{9}{4} sin (4 arcsin (\frac{1}{6} (- x + 6))) + C

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What is the integral of sqrt(x(6-x)) ?
The integral of sqrt(x(6-x)) is -9arcsin(sqrt(1/6 (-x+6)))+9/4 sin(4arcsin(sqrt(1/6 (-x+6))))+C