Encontrar la integral $$ \int \sin ^{2} heta \cos ^{2} heta d heta $$ Seleccione una: a. $$ \left(\frac{\sin ^{3} heta}{3} ight)\left(\frac{\cos ^{3} heta}{3} ight)+c $$ b. $$ \left( heta-\frac{1}{4} \sin 4 heta ight)+c $$ c. $$ \left(\frac{\sin ^{3} heta}{3} ight)+c $$ d. $$ \frac{1}{8}\left( heta-\frac{1}{4} \sin 4 heta ight)+c $$

Question

Encontrar la integral $$ \int \sin ^{2} 	heta \cos ^{2} 	heta d 	heta $$ Seleccione una: a. $$ \left(\frac{\sin ^{3} 	heta}{3}ight)\left(\frac{\cos ^{3} 	heta}{3}ight)+c $$ b. $$ \left(	heta-\frac{1}{4} \sin 4 	hetaight)+c $$ c. $$ \left(\frac{\sin ^{3} 	heta}{3}ight)+c $$ d. $$ \frac{1}{8}\left(	heta-\frac{1}{4} \sin 4 	hetaight)+c $$

UpStudy AI · Accepted Answer

Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \sin^{2}\left(\theta \right)\cos^{2}\left(\theta \right) d\theta$$
- step1: Multiply the terms:
  $$\int \left(\sin\left(\theta \right)\cos\left(\theta \right)\right)^{2} d\theta$$
- step2: Rewrite the expression:
  $$\int \frac{1}{4}\sin^{2}\left(2\theta \right) d\theta$$
- step3: Use properties of integrals:
  $$\frac{1}{4}\times \int \sin^{2}\left(2\theta \right) d\theta$$
- step4: Evaluate the integral:
  $$\frac{1}{4}\left(\frac{\theta }{2}-\frac{\sin\left(2\times 2\theta \right)}{4\times 2}\right)$$
- step5: Simplify:
  $$\frac{1}{4}\left(\frac{\theta }{2}-\frac{1}{8}\sin\left(4\theta \right)\right)$$
- step6: Use the distributive property:
  $$\frac{1}{4}\times \frac{\theta }{2}+\frac{1}{4}\left(-\frac{1}{8}\sin\left(4\theta \right)\right)$$
- step7: Multiply the terms:
  $$\frac{\theta }{8}+\frac{1}{4}\left(-\frac{1}{8}\sin\left(4\theta \right)\right)$$
- step8: Multiply the terms:
  $$\frac{\theta }{8}-\frac{1}{32}\sin\left(4\theta \right)$$
- step9: Add the constant of integral C:
  $$\frac{\theta }{8}-\frac{1}{32}\sin\left(4\theta \right) + C, C \in \mathbb{R}$$
Para resolver la integral $$ \int \sin^{2} \theta \cos^{2} \theta d \theta $$, podemos utilizar identidades trigonométricas.

Primero, recordemos que:

$$
\sin^{2} \theta \cos^{2} \theta = \frac{1}{4} \sin^{2}(2\theta)
$$

Por lo tanto, la integral se puede reescribir como:

$$
\int \sin^{2} \theta \cos^{2} \theta d \theta = \frac{1}{4} \int \sin^{2}(2\theta) d \theta
$$

Usando la identidad $$ \sin^{2}(x) = \frac{1 - \cos(2x)}{2} $$, tenemos:

$$
\int \sin^{2}(2\theta) d \theta = \int \frac{1 - \cos(4\theta)}{2} d \theta = \frac{1}{2} \int d\theta - \frac{1}{2} \int \cos(4\theta) d\theta
$$

Calculando cada parte:

1. $$ \int d\theta = \theta $$
2. $$ \int \cos(4\theta) d\theta = \frac{1}{4} \sin(4\theta) $$

Por lo tanto, la integral se convierte en:

$$
\frac{1}{2} \left( \theta - \frac{1}{4} \sin(4\theta) \right) = \frac{\theta}{2} - \frac{1}{8} \sin(4\theta)
$$

Finalmente, multiplicamos por $$ \frac{1}{4} $$:

$$
\frac{1}{4} \left( \frac{\theta}{2} - \frac{1}{8} \sin(4\theta) \right) = \frac{\theta}{8} - \frac{1}{32} \sin(4\theta) + C
$$

Por lo tanto, la solución de la integral es:

$$
\frac{\theta}{8} - \frac{1}{32} \sin(4\theta) + C
$$

Comparando con las opciones dadas, la respuesta correcta es:

**d. $$ \frac{1}{8}\left(\theta-\frac{1}{4} \sin 4 \theta\right)+c $$**.
La integral $$ \int \sin^{2} \theta \cos^{2} \theta d \theta $$ se resuelve utilizando identidades trigonométricas y se simplifica a $$ \frac{1}{8}\left(\theta-\frac{1}{4} \sin 4 \theta\right)+c $$.

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