First, simplifying the given's gives us  

And

Our goal is to get the given expression into terms of these two integrals. Our first step will be to try and get a  from our expression.

First note,

And for the third term,

Putting these facts together, we can rewrite the original expression as

Rearranging,

The three terms in parentheses can all be brought together, as the top limit of the previous integral matches the bottom limit of the next integral. Thus, we now have

Substituting in our given's, this simplifies to 

Here we are using several basic properties of definite integrals as well as the fundamental theorem of calculus. 

First, you can pull coefficients out to the front of integrals.

Second, we notice that our lower bound is bigger than our upper bound. You can switch the upper and lower bounds if you also switch the sign.

Lastly, our integral "distributes" over addition and subtraction. That means you can split the integral by each term and integrate each term separately. 

Now we integrate and calculate using the Fundamental Theorem of Calculus.

Rather than solve each integral independently, we can use the property of linearity to add the integrals together:

The integrals were solved using the following rule:

Finally, we evaluate the definite integral by evaluating the answer at the upper bound and subtracting the answer evaluated at the lower bound:

Begin by finding the velocity function by integrating the acceleration function.

We use  as the constant of integration since the function  is a velocity and at initially, at , the velocity is equals the constant of integration. 

At  seconds we are told the velocity is equal to . 
 

The average value of a function on a given interval  is given by the following function:

Now, let's simply input our values and function in:

The position function describing any object is the antiderivative of the velocity function (in other words, velocity is the derivative of position). 

So, we first integrate the velocity function:

The following rule was used for integration:

Now, to determine the constant of integration, we use our initial condition given,

Plugging this into our function, we get

Our final answer is

To find the position function from the acceleration function, we integrate the acceleration function to find the velocity function, and integrate again to get the position function:

The integral was found using the following rule:

To find the constant of integration, we use the initial velocity condition given:

Now, after replacing C with the known value, we integrate the velocity function to get the position function:

The same rule of integration was used as above.

We use the same procedure to solve for C, too, only this time using the initial position condition:

Our final answer is

We first must establish the following relationship

 and 

We now may note that 

 or 

Since 

We must plug in our initial condition 

Therefore our new velocity equation is 

We now may similarly integrate the velocity equation to find position.

Plugging in our second initial condition 

We find our final equation to be:

To find the velocity function, we integrate the acceleration function (the acceleration is the antiderivative of the velocity):

The rules of integration used were

, 

To solve for the integration constant, we plug in the given initial condition:

Our final answer is

To find the position function, we must integrate the velocity function, as velocity is the antiderivative of position:

The following rule of integration was used:

Finally, we use the initial condition to solve for the integration constant:

Our final answer is