Exercises

1. Find all antiderivatives of each of the following functions.

   `e^(-2t)`	`5t^(-1)`
   `t^(-3)`	`sin text[(]2 pi t text[)]`
   `1/t^(-2)`	`1/(2t+1)`
   `cos 3t`	`7/(3+2t)`

2. Find all antiderivatives of each of the following functions.

   `e^(-t) + sin 5t`	`e^(2x)-sin 3x`
   `e^(-2t) + cos 5t`	`x^2 -3^x+x^(3/2)`
   `t^2 -  cos 2t`	`5/(2+3x)-cos x`
   `3^(2t)`	`e^(2x)-1/(5x)+1/x^5`

3. Solve each of the following initial value problems (see Exercise 1).

   `(dy)/(dt)=e^(-2t)`, `ytext[(]0text[)]=1`	`(dy)/(dt)=5t^(-1)`, `ytext[(]1text[)]=1`
   `(dy)/(dt)=t^(-3)`, `ytext[(]1text[)]=0`	`(dy)/(dt)=sin text[(]2 pi t text[)]`, `ytext[(]0text[)]=1`
   `(dy)/(dt)=1/t^(-2)`, `ytext[(]1text[)]=0`	`(dy)/(dt)=1/(2t+1)`, `ytext[(]0text[)]=1`
   `(dy)/(dt)=cos 3t`, `ytext[(]0text[)]=1`	`(dy)/(dt)=7/(3+2t)`, `ytext[(]0text[)]=1`

4. Solve each of the following initial value problems (see Exercise 2).

   `(dy)/(dt)=e^(-t) + sin 5t`, `ytext[(]0text[)]=1`	`(dy)/(dx)=e^(2x)-sin 3x`, `ytext[(]0text[)]=1`
   `(dy)/(dt)=e^(-2t) + cos 5t`, `ytext[(]0text[)]=1`	`(dy)/(dx)=x^2 -3^x+x^(3/2)`, `ytext[(]0text[)]=1`
   `(dy)/(dt)=t^2 -  cos 2t`, `ytext[(]0text[)]=1`	`(dy)/(dx)=5/(2+3x)-cos x`, `ytext[(]0text[)]=1`
   `(dy)/(dt)=3^(2t)`, `ytext[(]0text[)]=1`	`(dy)/(dx)=e^(2x)-1/(5x)+1/x^5`, `ytext[(]1text[)]=0`

5. 1. Show that `-ln text[(] cos theta text[)]` is an antiderivative of `tan theta` for `-pi/2<theta<pi/2`.
   2. Use your graphing tool to graph both of the functions in part (a), and explain how features of each graph match those of the other.
6. Find all antiderivatives of each of the following functions.

   `1/(4+x^2)`	`1/(2+x^2)-3/sqrt(2-x^2)`
   `1/(1+4x^2)`	`x+1/x+1/x^2`
   `1/sqrt(4-x^2)`	`x+1+1/(x+1)+1/(x^2+1)`
   `1/sqrt(1-4x^2)`	`ln(e^x)`