integral de (x^2)/(sqrt(9-x^2))(1)

Solución

\int \frac{x ^{2}}{9 - x ^{2}} (1) d x

\frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2}) + C

Pasos de solución

\int \frac{x ^{2}}{9 - x ^{2}} \cdot 1 d x

Simplificar

= \int \frac{x ^{2}}{9 - x ^{2}} d x

Aplicar integración por sustitución trigonométrica

: \int 9 sin^{2} (u) d u

= \int 9 sin^{2} (u) d u

Sacar la constante:

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

= 9 \cdot \int sin^{2} (u) d u

Re-escribir usando identidades trigonométricas

= 9 \cdot \int \frac{1 - cos ( 2 u )}{2} d u

Sacar la constante:

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

= 9 \cdot \frac{1}{2} \cdot \int 1 - cos (2 u) d u

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

= 9 \cdot \frac{1}{2} (\int 1 d u - \int cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= 9 \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u))

Sustituir en la ecuación

u = arcsin (\frac{1}{3} x)

= 9 \cdot \frac{1}{2} (arcsin (\frac{1}{3} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{3} x)))

Simplificar

9 \cdot \frac{1}{2} (arcsin (\frac{1}{3} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) : \frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2})

= \frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2})

Agregar una constante a la solución

= \frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2}) + C

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