Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy. After evaluating each integral, write a short description of your method.

1. $\int x ^ { 2 } e ^ { - x } d x$

   Hint (a):

   Use integration by parts.
   Let $f = x^{2}$ and $dg = e^{–x}$.

2. $\int x \operatorname { sec } x ^ { 2 } d x$

   Hint (b):

   Use substitution.
   Let $u = x^{2}$.

   More Help (b):

   $\int\sec(x)dx=\ln|\sec(x)+\tan(x)|$

3. $\int _ { - \infty } ^ { \infty } \frac { 1 } { 1 + x ^ { 2 } } d x$

   Hint (c):

   You should recognize the integrand as the derivative of an inverse trigonometric function.

   More Help (c):

   This is an improper integral, so a limit will need to be used to evaluate it.
   The symmetry of the graph can also be used to evaluate the integral.

4. $\int _ { a } ^ { b } f ^ { \prime } ( x ) d x$

   Hint (d):

   Review the first Fundamental Theorem of Calculus.