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integral de sqrt(x)sqrt(81-x)

Solución

\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

Pasos de solución

\int x 81 - x d x

Aplicar las leyes de los exponentes:

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

x 81 - x = x (81 - x)

= \int x (81 - x) d x

Expandir

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

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

Completar el cuadrado

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

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

Aplicar integración por sustitución

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

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

Sacar la constante:

\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

Aplicar integración por sustitución trigonométrica

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

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

Sacar la constante:

\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

Re-escribir usando identidades trigonométricas

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

Sacar la constante:

\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

Aplicar la regla de la suma:

\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))

Sustitución hacia atrás

= \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}))))

Simplificar

\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)))

Agregar una constante a la solución

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

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