Substitution and Average Value of a function:

1. Evaluate each definite integral.

(a) $\displaystyle{ \int_{-1}^1  x(x^2+1)^2 \: dx}$  solution

(b)  $\displaystyle{ \int_0^1  x \sqrt{5x^2+4} \: \: dx}$  solution

(c)  $\displaystyle{ \int_1^2  \frac{x}{(x^2+3)^2} \: dx}$   solution 

2. Find the average value of the function $f$ over the indicated interval $[a,b]$.  solution

(a)  $f(x)=2x^2-3; \: \: [1, 3]$

(b)  $f(x)=x^2+2x -3; \: \: [-1, 2]$

(c)$f(x)=1+3x^2, \quad [-2, 1]$   solution

(d) $f(x)=\dfrac{10}{x+1} \quad [ 0, 4]$  solution