No calculator! Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method.

1. $\int \operatorname { cos } ^ { - 1 } ( x ) d x$ 

   Step 1(a):

   $f=\cos^{-1}(x), \ dg=dx, \ df=-\frac{1}{\sqrt{1-x^{2}}}dx, \ g=x$

   Step 2(a):

   $\int \cos^{-1}(x)dx=x\cos^{-1}(x)+\int \frac{xdx}{\sqrt{1-x^{2}}}$

   Hint (a):

   Use integration by parts and substitution.

2. $\int _ { \pi / 3 } ^ { \pi / 2 } \operatorname { csc } ( x ) \operatorname { cot } ( x ) d x$ 

   Hint (b):

   $\frac{d}{dx}\csc(x)=-\csc(x)\cot(x)$

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

   Hint (c):

   There is a vertical asymptote at $x = 0$, so this improper integral needs to be rewritten in proper form (as a limit).

   More help (c):

   Write a sum of two proper integrals.

4. $\int \operatorname { sec } ^ { 2 } ( x ) \operatorname { tan } ( x ) d x$

   Hint (d):

   Use $u$-substitution. $\sec^2(x)\tan(x) = (\sec(x))(\sec(x))(\tan(x))$