integral de 1/(sqrt((x^2-a^2)^3))

Solución

\int \frac{1}{( x ^{2} - a ^{2} ) ^{3}} d x

- \frac{x}{a ^{2} x ^{2} - a ^{2}} + C

Pasos de solución

\int \frac{1}{( x ^{2} - a ^{2} ) ^{3}} d x

(x^{2} - a^{2})^{3} = (x^{2} - a^{2})^{\frac{3}{2}},

asumiendo que

(x^{2} - a^{2}) \geq 0

= \int \frac{1}{( x ^{2} - a ^{2} ) ^{\frac{3}{2}}} d x

Aplicar integración por sustitución trigonométrica

: \int \frac{sec ( u )}{a ^{2} tan ^{2} ( u )} d u

= \int \frac{sec ( u )}{a ^{2} tan ^{2} ( u )} d u

Sacar la constante:

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

= \frac{1}{a ^{2}} \cdot \int \frac{sec ( u )}{tan ^{2} ( u )} d u

Re-escribir usando identidades trigonométricas

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

Aplicar integración por sustitución

: \int \frac{1}{v ^{2}} d v

= \frac{1}{a ^{2}} \cdot \int \frac{1}{v ^{2}} d v

Aplicar la regla de la potencia

: - \frac{1}{v}

= \frac{1}{a ^{2}} (- \frac{1}{v})

Sustitución hacia atrás

= \frac{1}{a ^{2}} (- \frac{1}{sin ( arcsec ( \frac{1}{a} x ) )})

Simplificar

\frac{1}{a ^{2}} (- \frac{1}{sin ( arcsec ( \frac{1}{a} x ) )}) : - \frac{x}{a ^{2} x ^{2} - a ^{2}}

= - \frac{x}{a ^{2} x ^{2} - a ^{2}}

Agregar una constante a la solución

= - \frac{x}{a ^{2} x ^{2} - a ^{2}} + C

\int - 0.08 x^{2} + 4 x d x

\int x^{2} (4 - x^{3})^{4} d x

\int (x^{2} + x) (e^{1 - 2 x}) d x

\int \frac{1}{cos ^{2} ( x ) + sin ^{2} ( x )} d x