Part 1

1. Integrate and simplify your result.
   
   $$\int_{0}^{\pi/3} \cos^4 t \sin^2 t dt$$
   
   

   Solution: $\pi/48+\sqrt{3}/64$.

2. Integrate
   
   $$\int \frac{dx}{x^2\sqrt{9-x^2} }$$
   
   

   Solution: $-\frac{\sqrt{9-x^2}}{9x}+C$

3. Integrate
   
   $$\int \frac{1}{x^2(x^2+4)} dx$$
   
   

   Solution: $-\frac{1}{4x} -\frac{1}{8} \arctan \frac{x}{2} +C$

4. Determine if the following integral converges or diverges, and evaluate the integral if it converges.
   
   $$\int_{0}^{\infty} x^2 e^{-x^3} dx$$
   
   

   Solution: Converges, to $\frac{1}{3}$