integral de sqrt(2-3x^2)

Solución

\int 2 - 3 x^{2} d x

\frac{1}{2 3} (2 arcsin (\frac{3}{2} x) + sin (2 arcsin (\frac{3}{2} x))) + C

Pasos de solución

\int 2 - 3 x^{2} d x

Aplicar integración por sustitución trigonométrica

: \int \frac{2}{3} cos^{2} (u) d u

= \int \frac{2}{3} cos^{2} (u) d u

Sacar la constante:

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

= \frac{2}{3} \cdot \int cos^{2} (u) d u

Re-escribir usando identidades trigonométricas

= \frac{2}{3} \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

= \frac{2}{3} \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

= \frac{2}{3} \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)

= \frac{2}{3} \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Sustituir en la ecuación

u = arcsin (\frac{3}{2} x)

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

Simplificar

\frac{2}{3} \cdot \frac{1}{2} (arcsin (\frac{3}{2} x) + \frac{1}{2} sin (2 arcsin (\frac{3}{2} x))) : \frac{1}{2 3} (2 arcsin (\frac{3}{2} x) + sin (2 arcsin (\frac{3}{2} x)))

= \frac{1}{2 3} (2 arcsin (\frac{3}{2} x) + sin (2 arcsin (\frac{3}{2} x)))

Agregar una constante a la solución

= \frac{1}{2 3} (2 arcsin (\frac{3}{2} x) + sin (2 arcsin (\frac{3}{2} x))) + C

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