integral-3sec(x)*sec^3(x)

Solución

integral

- 3 sec (x) \cdot sec^{3} (x)

- 3 tan (x) - tan^{3} (x) + C

Pasos de solución

\int - 3 sec (x) sec^{3} (x) d x

Sacar la constante:

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

= - 3 \cdot \int sec (x) sec^{3} (x) d x

Simplificar

sec (x) sec^{3} (x) : sec^{4} (x)

= - 3 \cdot \int sec^{4} (x) d x

Re-escribir usando identidades trigonométricas

= - 3 \cdot \int (1 + tan^{2} (x)) sec^{2} (x) d x

Aplicar integración por sustitución

: \int 1 + u^{2} d u

= - 3 \cdot \int 1 + u^{2} 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

= - 3 (\int 1 d u + \int u^{2} d u)

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= - 3 (u + \frac{u ^{3}}{3})

Sustituir en la ecuación

u = tan (x)

= - 3 (tan (x) + \frac{tan ^{3} ( x )}{3})

Simplificar

- 3 (tan (x) + \frac{tan ^{3} ( x )}{3}) : - 3 tan (x) - tan^{3} (x)

= - 3 tan (x) - tan^{3} (x)

Agregar una constante a la solución

= - 3 tan (x) - tan^{3} (x) + C

- sec^{2} (x)

- 3 sec (x) \cdot (sec (x))^{3}

\frac{1}{( s + a )}

- 3 v + 5

3 cos^{2} (x)