integral de 1/(sin^2(x/2)cos^2(x/2))

Solución

\int \frac{1}{sin ^{2} ( \frac{x}{2} ) cos ^{2} ( \frac{x}{2} )} d x

- 4 cot (x) + C

Pasos de solución

\int \frac{1}{sin ^{2} ( \frac{x}{2} ) cos ^{2} ( \frac{x}{2} )} d x

Re-escribir usando identidades trigonométricas

= \int \frac{8}{1 - cos ( 2 x )} d x

Sacar la constante:

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

= 8 \cdot \int \frac{1}{1 - cos ( 2 x )} d x

Re-escribir usando identidades trigonométricas

= 8 \cdot \int \frac{1}{2 sin ^{2} ( x )} d x

Sacar la constante:

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

= 8 \cdot \frac{1}{2} \cdot \int \frac{1}{sin ^{2} ( x )} d x

Aplicar la regla de integración:

\int \frac{1}{sin ^{2} ( x )} d x = - cot (x)

= 8 \cdot \frac{1}{2} (- cot (x))

Simplificar

8 \cdot \frac{1}{2} (- cot (x)) : - 4 cot (x)

= - 4 cot (x)

Agregar una constante a la solución

= - 4 cot (x) + C

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