No calculator! Evaluate each definite integral below. If the integral is improper, rewrite it in “proper” limit notation first. One of these integrals will need to be written as the sum of two improper integrals as the first step.

1. $\int _ { 1 } ^ { \infty } \frac { \operatorname { tan } ^ { - 1 } ( x ) d x } { 1 + x ^ { 2 } }$ 

   Hint (a):

   Let $U = \tan^{−1}(x)$.

2. $\int _ { 0 } ^ { \pi } x \operatorname { cos } ( x ) d x$ 

   Step 1(b):

   Use integration by parts;
   If $f(x) = x$ and $dg = \cos(x)dx$,
   then $df = \text{___}$ and $g(x) = \text{____}$.

   Step 2(b):

   Evaluate and solve.

   $=f(x)g(x)\Big|_0^\pi-\int_{0}^{\pi }g(x)df$

3. $\int _ { 0 } ^ { 4 } \frac { d x } { ( x - 2 ) ^ { 2 / 3 } }$ 

   Hint (c):

   Write this integral as the sum of two improper integrals.

   Step (c):

   $\lim \limits_{a\rightarrow 2^{-}}\int_{0}^{a}\frac{dx}{(x-2 )^{\frac{2}{3}}}+\lim \limits_{a\rightarrow 2^{+}}\int_{a}^{4}\frac{dx}{(x-2)^{\frac{2}{3}}}$