integral de 1/([(ax)^2-b^2]^{3/2)}

Solución

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

- \frac{x}{b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{1}{2}}} + C

Pasos de solución

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

Aplicar integración por sustitución

: \int \frac{1}{a ( u ^{2} - b ^{2} ) ^{\frac{3}{2}}} d u

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

Sacar la constante:

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

= \frac{1}{a} \cdot \int \frac{1}{( u ^{2} - b ^{2} ) ^{\frac{3}{2}}} d u

Aplicar integración por sustitución trigonométrica

: \int \frac{sec ( v )}{b ^{2} tan ^{2} ( v )} d v

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

Sacar la constante:

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

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

Re-escribir usando identidades trigonométricas

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

Aplicar integración por sustitución

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

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

Aplicar la regla de la potencia

: - \frac{1}{w}

= \frac{1}{a} \cdot \frac{1}{b ^{2}} (- \frac{1}{w})

Sustitución hacia atrás

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

Simplificar

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

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

Agregar una constante a la solución

= - \frac{x}{b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{1}{2}}} + C

\frac{d}{d x} (7 x (x^{3} - 8 x + 5))

x \to 0 lim (\frac{e ^{x} - 1}{2 x})

y = \frac{2}{x}, (36, \frac{2}{6})

\frac{d}{d x} (- x + \frac{6}{x} + 1)

\int \frac{2 x + 3}{x ^{3} + x} d x