Position can be found by integrating velocity with respect to time:

For velocities:

The position functions are:

These constants of integration can be found by using the given initial conditions:

Which gives the definite integrals:

Position can be found by integrating with respect to time velocity once, or integrating with respect to time acceleration twice:

Begin by finding velocity functions. For the acceleration functions

Velocities are:

Find the constants of integration by using the given initial conditions:

Plugging these values back in gives the definite velocity functions:

Now, find the position functions by integrating once more:

Find these new constants of integration the same way as before:

And the definite position functions are thus:

Then at time , the particle's position is:

To find the position function for the race car, we must integrate the velocity function.

To integrate this function, we must use the following formula:

Now, to find the position at time , we plug in  into the position function.

To find when it will get back to Tom. Assume Tom is at position 0, and find where the position function equals zero. Position funtion is the integral of the velocity funtion which is the intergral of the acceleration function. 

using, 

.

 by what's given.

 since we assume Tom is at position 0, so now we must find where it is 0 again.

Thus, it will return to Tom at posistion after 5 seconds.

To find when the particle's position has maxima and minima, take the derivative of the position function with respect to time to find velocity, and find the times when this velocity is equal to zero:

For the position function

The velocity function is

This function is equal to zero at time . Since it is positive prior and negative afterwards, this indicates a maximum position.

Now, to find the time when the particle is first halfway to this position, use the equation;

The solutions to the equation are 

So the time when the particle reaches half of its maximum position is therefore:

The position function can be found by integrating the velocity function with respect to time. For the velocity function

Exponential and sin functions integrate as follows:

Note that after integration of a function, we always include a constant of integration, usually denoted as . Furthermore, if multiple functions are added/multiplied together, a single integration only requires one constant of integration, even though there are multiple terms.

Using this knowledge, the position function can be found 

The constant of integration can be found via knowledge of the initial position:

This gives the definite integral

At time 

Position can be found by integrating the acceleration function twice, though it's usually useful to find velocity by integrating acceleration once, and then integrating the velocity to find position, so that we may deal with constants of integration as they appear.

For the acceleration function

The integral is found to be

Upon integrating a function, we must always add a constant of integration. Note that the integral of a constant  will be found to be 

We're told the rock is dropped, and may assume a zero initial velocity:

Now, the position function can be found by integrating once more, noting the property that :

Recall that we were given an initial height:

At time 

The kinematic equations in physics are actually derived in this manner, presuming a constant acceleration.

Position can be found by integrating velocity with respect to time. For velocity functions:

The position functions are

These can be found via the properties 

And  is a constant of integration.

These constants of integration can be found by using the initial conditions given, namely

; 

; 

; 

So the definite integrals are tus:

At at time 

; 

; 

; 

Position can be found by integrating the velocity function with respect to time:

We are given the velocity function

To integrate this, it is necessary to know how to integrate exponential functions:

The velocity function has a sum of exponential functions, and for sums, each element can be integrated independently:

(Note that for a single integration, only one constant of integration is needed regardless of the number of terms)

To find the constant of integration, utilize the initial condition:

Now to find the farthest potential position, take the limit of this function as t approaches infinity:

Since the class of possible position functions is given by the antiderivative of the velocity function:

=

Now, impose the initial condition:

so:

and

Note that  is equivalent to  in radians because they differ by an integer multiple of . So we have