$($a$)$ Use the substitution $u=1+x^6$ to compute ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^5}{1+x^6}dx.$

$($b$)$ Use the substitution $u=x^3$ to compute ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^2}{1+x^6}dx.$

Hint: Remember the derivative of $arctan(u).$

$($c$)$ Find two numbers $m$ and $n$ so that ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^m}{1+x^{10}}dx$ can be computed using a strategy very similar to part $($a$),$ and ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^n}{1+x^{10}}dx$ can be computed using a strategy very similar to part $($b$).$ Explain your reasoning.

$2.$ A car is driving along a road. At time $t,$ measured in minutes past $12$:$00,$ its velocity is $f(t)=1200+900t$ feet per minute.

$($a$)$ Find the distance traveled by the car from $12$:$00$ to $12$:$04,$ by writing down an integral and then computing it$.$ Your answer should be fully simplified with correct units.

$($b$)$ Let $g(s)$ be the car's velocity in feet per second at time $s,$ where $s$ is the number of seconds past $12$:$00.$ Do a unit conversion of the formula $f(t)=1200+900t$ to determine the formula for $g(s).$ Hint: Since $t$ is the number of minutes past $12$:$00$ while $s$ is the number of seconds past $12$:$00,$ it is true that $s=60t($and $t=\frac{s}{60}).$ Taking $f(t)=1200+900t$ and substituting $t=\frac{s}{60}$ gives the velocity in feet per minute at time $s$ seconds past $12$:$00.$ There is one more step to get the formula for $g(s).$

$($c$)$ Use your $g(s)$ formula from part $($b$)$ to write down a $ds$ integral for the distance traveled by the car from $12$:$00$ to $12$:$04.($Hint: What value of $s$ corresponds to the time $12$:$04?)$ Compute the integral and check that your answer is the same as in part $($a$).$

$($d$)$ Use the substitution $s=60t$ to convert your $dt$ integral from part $($a$)$

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into a $ds$ integral. Make sure also to change the limits of integration. Check that the resulting $ds$ integral is the same as the one you computed in part $($c$).$ The lesson is that the process of substituting $s=60t$ effectively changes the time units from minutes to seconds in the problem.

$3.$ In this problem you will compute the integral ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}2sin(x)cos(x)dx$ in three different ways.

$($a$)$ Compute the integral using the substitution $u=sin(x).$

$($b$)$ Compute the integral using the substitution $u=cos(x).$

$($c$)$ Compute the integral by applying the trig identity $2sin(x)cos(x)=sin(2x)$ and then using the substitution $u=2x.$

$($d$)$ Use the linked graph to explain why all three answers to parts $($a$),($b$),($c$)$ are correct even though they look different.

$($e$)$ Compute ${\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}2sin(x)cos(x)dx$ three different times, using your answers to parts $($a$),($b$),$ and $($c$),$ and check that all three computations give the same result.