Evaluate the following integrals without using your calculator. Be sure to change the bounds if you use $u$-substitution.

1. $\int _ { 1 } ^ { 5 } \frac { x } { \sqrt { x ^ { 2 } - 1 } } d x$ 

   Steps (a):

   Let $u = x^2 − 1$.
   Then $du = 2x dx$.

   Rewrite the integrand and do not forget to write the bounds in terms of $u$.

   Answer (a):

   $=\int_0^{24}\frac{1}{2\sqrt{u}}=u^{1/2}\Big|_0^{24}=?$

2. $\int _ { 0 } ^ { 1 } x \sqrt { 1 - x ^ { 2 } } d x$ 

   Hint (b):

   Let $u = 1 − x^2$.

3. $\int _ { 0 } ^ { 2 } \frac { x } { x ^ { 2 } + 1 } d x$ 

   Answer (c):

   $=\int_1^5\frac{1}{2u}du=\frac{1}{2}\ln(u)\Big|_1^5=\frac{1}{2}\ln(5)$

4. $\int _ { 0 } ^ { 2 } 3 x e ^ { - x ^ { 2 } } d x$ 

   Answer (d):

   $=\int_0^{-4}-\frac{3}{2}e^udu=\frac{3}{2}-\frac{3}{2}e^{-4}$