What is the integral of sqrt(x)sqrt(81-x) ?

The integral of sqrt(x)sqrt(81-x) is 6561/16 (2arcsin(2/81 x-1)+sin(2arcsin(2/81 x-1)))+C

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integral of sqrt(x)sqrt(81-x)

\int x 81 - x d x

\frac{6561}{16} (2 arcsin (\frac{2}{81} x - 1) + sin (2 arcsin (\frac{2}{81} x - 1))) + C

Solution steps

\int x 81 - x d x

Apply exponent rule:

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

x 81 - x = x (81 - x)

= \int x (81 - x) d x

Expand

x (81 - x) : 81 x - x^{2}

= \int 81 x - x^{2} d x

Complete the square

81 x - x^{2} : - (x - \frac{81}{2})^{2} + \frac{6561}{4}

= \int - (x - \frac{81}{2})^{2} + \frac{6561}{4} d x

Apply u-substitution

: \int \frac{- 4 u ^{2} + 6561}{2} d u

= \int \frac{- 4 u ^{2} + 6561}{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} + 6561 d u

Apply Trigonometric Substitution

: \int \frac{6561}{2} cos^{2} (v) d v

= \frac{1}{2} \cdot \int \frac{6561}{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{6561}{2} \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= \frac{1}{2} \cdot \frac{6561}{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{6561}{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{6561}{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{6561}{2} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= \frac{1}{2} \cdot \frac{6561}{2} \cdot \frac{1}{2} (arcsin (\frac{2}{81} (x - \frac{81}{2})) + \frac{1}{2} sin (2 arcsin (\frac{2}{81} (x - \frac{81}{2}))))

Simplify

\frac{1}{2} \cdot \frac{6561}{2} \cdot \frac{1}{2} (arcsin (\frac{2}{81} (x - \frac{81}{2})) + \frac{1}{2} sin (2 arcsin (\frac{2}{81} (x - \frac{81}{2})))) : \frac{6561}{16} (2 arcsin (\frac{2}{81} x - 1) + sin (2 arcsin (\frac{2}{81} x - 1)))

= \frac{6561}{16} (2 arcsin (\frac{2}{81} x - 1) + sin (2 arcsin (\frac{2}{81} x - 1)))

Add a constant to the solution

= \frac{6561}{16} (2 arcsin (\frac{2}{81} x - 1) + sin (2 arcsin (\frac{2}{81} x - 1))) + C

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What is the integral of sqrt(x)sqrt(81-x) ?
The integral of sqrt(x)sqrt(81-x) is 6561/16 (2arcsin(2/81 x-1)+sin(2arcsin(2/81 x-1)))+C