${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}cosx-sinxdx$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}xcos(-3x^2+4)dx$

${\displaystyle \underset{0\mathrm{.}5}{\overset{\frac{+1}{2}}{}}}sin(2x-1)dx$

$3.$Derive the formula for ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}cotxdx=ln|sinx|+C.$

$($Hint: Use the fact that $cotx=\frac{cosx}{sinx}$ and integrate using substitution. Check the notes for the example with tangent if you need.$)$

$4.$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(2x+1)sin(2x-1)dx$

$5.$The following model gives the flux of radio emissions from the sun: $1\mathrm{.}48,$ where $t$ is the number of years since January $1,1997,$ and $F(t)$ is the flux of solar emissions at time $t.$

Estimate the average number of sunspots visible over the $5$ year period beginning January $1,2001.$ Remember that the average of a function over the interval $a,b$ is $\frac{1}{b-a}{\displaystyle \underset{a}{\overset{b}{}}}f(x)dx.$