Exercise 10.3: Importance sampling

1. Choose the weight function $P(x)=e{-x}$ and evaluate the integral:
   
   $$\int _0^{\infty} {x^{3/2}e^{-x}dx}.$$
   
   

2. Choose $P(x)=e^{-ax}$ and estimate the integral
   
   $$\int _0^{\pi} \frac{dx}{x^2+\cos ^2{x}}.$$
   
   
   Determine the value of $a$ that minimizes the variance of the integral.