integral de (cos^3(5x))/(sin^4(5x))

Solución

\int \frac{cos ^{3} ( 5 x )}{sin ^{4} ( 5 x )} d x

\frac{1}{5} (- \frac{1}{3} csc^{3} (5 x) + csc (5 x)) + C

Pasos de solución

\int \frac{cos ^{3} ( 5 x )}{sin ^{4} ( 5 x )} d x

Aplicar integración por sustitución

: \int \frac{cos ^{3} ( u )}{5 sin ^{4} ( u )} d u

= \int \frac{cos ^{3} ( u )}{5 sin ^{4} ( u )} d u

Sacar la constante:

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

= \frac{1}{5} \cdot \int \frac{cos ^{3} ( u )}{sin ^{4} ( u )} d u

Re-escribir usando identidades trigonométricas

= \frac{1}{5} \cdot \int \frac{( 1 - sin ^{2} ( u ) ) cos ( u )}{sin ^{4} ( u )} d u

Aplicar integración por sustitución

: \int \frac{1 - v ^{2}}{v ^{4}} d v

= \frac{1}{5} \cdot \int \frac{1 - v ^{2}}{v ^{4}} d v

Expandir

\frac{1 - v ^{2}}{v ^{4}} : \frac{1}{v ^{4}} - \frac{1}{v ^{2}}

= \frac{1}{5} \cdot \int \frac{1}{v ^{4}} - \frac{1}{v ^{2}} d v

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}{5} (\int \frac{1}{v ^{4}} d v - \int \frac{1}{v ^{2}} d v)

\int \frac{1}{v ^{4}} d v = - \frac{1}{3 v ^{3}}

\int \frac{1}{v ^{2}} d v = - \frac{1}{v}

= \frac{1}{5} (- \frac{1}{3 v ^{3}} - (- \frac{1}{v}))

Sustitución hacia atrás

= \frac{1}{5} (- \frac{1}{3 sin ^{3} ( 5 x )} - (- \frac{1}{sin ( 5 x )}))

Simplificar

\frac{1}{5} (- \frac{1}{3 sin ^{3} ( 5 x )} - (- \frac{1}{sin ( 5 x )})) : \frac{1}{5} (- \frac{1}{3} csc^{3} (5 x) + csc (5 x))

= \frac{1}{5} (- \frac{1}{3} csc^{3} (5 x) + csc (5 x))

Agregar una constante a la solución

= \frac{1}{5} (- \frac{1}{3} csc^{3} (5 x) + csc (5 x)) + C

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