xiv) $$ \int \frac{6 \cos t}{(2+\operatorname{sen} t)^{3}} d t \quad $$ xv) $$ \int \sqrt{\frac{x-1}{x^{5}}} d x \quad $$ xvi) $$ \int \frac{t}{t^{2}+4} d t $$ xvii) $$ \int \frac{1}{x^{2}+4} d x \quad $$ xviii) $$ \int \frac{1}{\sqrt{4-x^{2}}} d x $$

Question

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Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{6\cos\left(t\right)}{\left(2+\sin\left(t\right)\right)^{3}} dt$$
- step1: Rewrite the expression:
  $$\int 6\times \frac{\cos\left(t\right)}{\left(2+\sin\left(t\right)\right)^{3}} dt$$
- step2: Use properties of integrals:
  $$6\times \int \frac{\cos\left(t\right)}{\left(2+\sin\left(t\right)\right)^{3}} dt$$
- step3: Use the substitution $$dt=\frac{1}{\cos\left(t\right)} dv$$ to transform the integral$$:$$
  $$6\times \int \frac{\cos\left(t\right)}{\left(2+\sin\left(t\right)\right)^{3}}\times \frac{1}{\cos\left(t\right)} dv$$
- step4: Simplify:
  $$6\times \int \frac{1}{\left(2+\sin\left(t\right)\right)^{3}} dv$$
- step5: Use the substitution $$v=\sin\left(t\right)$$ to transform the integral$$:$$
  $$6\times \int \frac{1}{\left(2+v\right)^{3}} dv$$
- step6: Use the substitution $$dv=1 dt$$ to transform the integral$$:$$
  $$6\times \int \frac{1}{\left(2+v\right)^{3}}\times 1 dt$$
- step7: Simplify:
  $$6\times \int \frac{1}{\left(2+v\right)^{3}} dt$$
- step8: Use the substitution $$t=2+v$$ to transform the integral$$:$$
  $$6\times \int \frac{1}{t^{3}} dt$$
- step9: Evaluate the integral:
  $$6\times \frac{t^{-3+1}}{-3+1}$$
- step10: Simplify:
  $$6\times \frac{t^{-2}}{-2}$$
- step11: Rewrite the expression:
  $$6\left(-\frac{t^{-2}}{2}\right)$$
- step12: Multiply the terms:
  $$-6\times \frac{t^{-2}}{2}$$
- step13: Reduce the fraction:
  $$-3t^{-2}$$
- step14: Substitute back:
  $$-3\left(2+v\right)^{-2}$$
- step15: Simplify:
  $$-\frac{3}{\left(2+v\right)^{2}}$$
- step16: Substitute back:
  $$-\frac{3}{\left(2+\sin\left(t\right)\right)^{2}}$$
- step17: Add the constant of integral C:
  $$-\frac{3}{\left(2+\sin\left(t\right)\right)^{2}} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{1}{\sqrt{4-x^{2}}} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \frac{1}{\sqrt{4-x^{2}}} dx$$
- step1: Evaluate the integral:
  $$\arcsin\left(\frac{x}{2}\right)$$
- step2: Add the constant of integral C:
  $$\arcsin\left(\frac{x}{2}\right) + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{t}{t^{2}+4} d t $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{t}{t^{2}+4} dt$$
- step1: Use the substitution $$dt=\frac{1}{2t} dv$$ to transform the integral$$:$$
  $$\int \frac{t}{t^{2}+4}\times \frac{1}{2t} dv$$
- step2: Simplify:
  $$\int \frac{1}{2t^{2}+8} dv$$
- step3: Use the substitution $$v=t^{2}$$ to transform the integral$$:$$
  $$\int \frac{1}{2v+8} dv$$
- step4: Rewrite the expression:
  $$\int \frac{1}{2}\times \frac{1}{v+4} dv$$
- step5: Use properties of integrals:
  $$\frac{1}{2}\times \int \frac{1}{v+4} dv$$
- step6: Evaluate the integral:
  $$\frac{1}{2}\ln{\left(\left|v+4\right|\right)}$$
- step7: Substitute back:
  $$\frac{1}{2}\ln{\left(\left|t^{2}+4\right|\right)}$$
- step8: Calculate the absolute value:
  $$\frac{1}{2}\ln{\left(t^{2}+4\right)}$$
- step9: Add the constant of integral C:
  $$\frac{1}{2}\ln{\left(t^{2}+4\right)} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{1}{x^{2}+4} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \frac{1}{x^{2}+4} dx$$
- step1: Evaluate the integral:
  $$\frac{1}{2}\arctan\left(\frac{x}{2}\right)$$
- step2: Add the constant of integral C:
  $$\frac{1}{2}\arctan\left(\frac{x}{2}\right) + C, C \in \mathbb{R}$$
Aquí están las soluciones a las integrales solicitadas:

1. **Integral** $$ \int \frac{6 \cos t}{(2+\operatorname{sen} t)^{3}} d t $$:
   $$
   -\frac{3}{(2+\sin(t))^{2}} + C, \quad C \in \mathbb{R}
   $$

2. **Integral** $$ \int \sqrt{\frac{x-1}{x^{5}}} d x $$:
   - Esta integral excedió el tiempo de cálculo permitido, por lo que no se pudo resolver en este momento.

3. **Integral** $$ \int \frac{t}{t^{2}+4} d t $$:
   $$
   \frac{1}{2}\ln{(t^{2}+4)} + C, \quad C \in \mathbb{R}
   $$

4. **Integral** $$ \int \frac{1}{x^{2}+4} d x $$:
   $$
   \frac{1}{2}\arctan\left(\frac{x}{2}\right) + C, \quad C \in \mathbb{R}
   $$

5. **Integral** $$ \int \frac{1}{\sqrt{4-x^{2}}} d x $$:
   $$
   \arcsin\left(\frac{x}{2}\right) + C, \quad C \in \mathbb{R}
   $$

Si necesitas ayuda adicional con la integral que no se pudo calcular, házmelo saber.
Aquí están las soluciones a las integrales: 1. $$ \int \frac{6 \cos t}{(2+\sin t)^3} dt = -\frac{3}{(2+\sin t)^2} + C $$ 2. $$ \int \sqrt{\frac{x-1}{x^5}} dx $$ no se pudo resolver en este momento. 3. $$ \int \frac{t}{t^2+4} dt = \frac{1}{2} \ln(t^2+4) + C $$ 4. $$ \int \frac{1}{x^2+4} dx = \frac{1}{2} \arctan\left(\frac{x}{2}\right) + C $$ 5. $$ \int \frac{1}{\sqrt{4-x^2}} dx = \arcsin\left(\frac{x}{2}\right) + C $$ Si necesitas ayuda con la integral que no se pudo resolver, avísame.

xiv) \( \int \frac{6 \cos t}{(2+\operatorname{sen} t)^{3}} d t \quad \) xv) \( \int \sqrt{\frac{x-1}{x^{5}}} d x \quad \) xvi) \( \int \frac{t}{t^{2}+4} d t \) xvii) \( \int \frac{1}{x^{2}+4} d x \quad \) xviii) \( \int \frac{1}{\sqrt{4-x^{2}}} d x \)

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