integral de 1/(csc^4(3x)sec^2(3x))

Solución

\int \frac{1}{csc ^{4} ( 3 x ) sec ^{2} ( 3 x )} d x

\frac{1}{288} (18 x - 3 sin (6 x) - 4 sin^{3} (3 x) cos (3 x) + 16 sin^{5} (3 x) cos (3 x)) + C

Pasos de solución

\int \frac{1}{csc ^{4} ( 3 x ) sec ^{2} ( 3 x )} d x

Re-escribir usando identidades trigonométricas

= \int sin^{4} (3 x) cos^{2} (3 x) d x

Aplicar integración por sustitución

: \int sin^{4} (u) cos^{2} (u) \frac{1}{3} d u

= \int sin^{4} (u) cos^{2} (u) \frac{1}{3} d u

Sacar la constante:

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

= \frac{1}{3} \cdot \int sin^{4} (u) cos^{2} (u) d u

Re-escribir usando identidades trigonométricas

= \frac{1}{3} \cdot \int sin^{4} (u) (1 - sin^{2} (u)) d u

Expandir

sin^{4} (u) (1 - sin^{2} (u)) : sin^{4} (u) - sin^{6} (u)

= \frac{1}{3} \cdot \int sin^{4} (u) - sin^{6} (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{1}{3} (\int sin^{4} (u) d u - \int sin^{6} (u) d u)

\int sin^{4} (u) d u = - \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))

\int sin^{6} (u) d u = - \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)))

= \frac{1}{3} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)) - (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)))))

Sustituir en la ecuación

u = 3 x

= \frac{1}{3} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x)) - (- \frac{cos ( 3 x ) sin ^{5} ( 3 x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x)))))

Simplificar

\frac{1}{3} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x)) - (- \frac{cos ( 3 x ) sin ^{5} ( 3 x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x))))) : \frac{1}{288} (18 x - 3 sin (6 x) - 4 sin^{3} (3 x) cos (3 x) + 16 sin^{5} (3 x) cos (3 x))

= \frac{1}{288} (18 x - 3 sin (6 x) - 4 sin^{3} (3 x) cos (3 x) + 16 sin^{5} (3 x) cos (3 x))

Agregar una constante a la solución

= \frac{1}{288} (18 x - 3 sin (6 x) - 4 sin^{3} (3 x) cos (3 x) + 16 sin^{5} (3 x) cos (3 x)) + C

\frac{\partial}{\partial y} (3 x y e^{- x^{2} - y^{2}})

\int sin (5 x) sin (2 x) d x

\frac{\partial}{\partial x} (7 x y^{2})

x \to - 3 lim (- 4 x + 6)

\int cot^{4} (3 x) d x