integral de (sin^2(2x))/(csc^3(x))

Solución

\int \frac{sin ^{2} ( 2 x )}{csc ^{3} ( x )} d x

4 (- \frac{cos ^{3} ( x )}{3} + \frac{2 cos ^{5} ( x )}{5} - \frac{cos ^{7} ( x )}{7}) + C

Pasos de solución

\int \frac{sin ^{2} ( 2 x )}{csc ^{3} ( x )} d x

Utilizar la identidad trigonométrica básica:

\frac{1}{csc ( x )} = sin (x)

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

Re-escribir usando identidades trigonométricas

= \int 4 sin^{5} (x) cos^{2} (x) d x

Sacar la constante:

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

= 4 \cdot \int sin^{5} (x) cos^{2} (x) d x

Re-escribir usando identidades trigonométricas

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

Aplicar integración por sustitución

: \int - u^{2} (1 - u^{2})^{2} d u

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

Expandir

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

= 4 \cdot \int - u^{2} + 2 u^{4} - u^{6} 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

= 4 (- \int u^{2} d u + \int 2 u^{4} d u - \int u^{6} d u)

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

\int 2 u^{4} d u = \frac{2 u ^{5}}{5}

\int u^{6} d u = \frac{u ^{7}}{7}

= 4 (- \frac{u ^{3}}{3} + \frac{2 u ^{5}}{5} - \frac{u ^{7}}{7})

Sustituir en la ecuación

u = cos (x)

= 4 (- \frac{cos ^{3} ( x )}{3} + \frac{2 cos ^{5} ( x )}{5} - \frac{cos ^{7} ( x )}{7})

Agregar una constante a la solución

= 4 (- \frac{cos ^{3} ( x )}{3} + \frac{2 cos ^{5} ( x )}{5} - \frac{cos ^{7} ( x )}{7}) + C

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