We know that the derivative of  is .

So substituting

 allows us to have 

.

This allows us to rewrite the integral as 

 which, when integrated, gives us 

.

Substiting x back in gives us the answer, 

.

To evaluate the integral, we must recognize that what we were given looks very similar to the following integral:

To make our integral look like the one above, we must perform the following substiution:

Now, rewrite our integral:

It looks like the one above, so we can integrate now:

Finally, replace u with our original term:

To integrate, we must break the integral into three integrals:

The first integral is equal to 

and was found using the following rule:

The second integral is equal to

and was found using the following rule:

The final integral is found by performing the following substitution:

Now, rewrite and integrate:

The integral was found using the following rule:

Finally, rewrite the integral in terms of  by replacing  with the original term, and add all three integrals together to get a final answer of 

To integrate, we must make the following substitution:

The derivative was found using the following rule:

Now, rewrite the integral:

Notice that we changed

Next, distribute and integrate:

The integral was found using the following rule:

Finally, replace  with our original  term:

To solve the integral, we must make the following substitution:

Now, rewrite the integral and integrate:

The integral was found using the following rule:

Finally, replace  with our original  term:

To evaluate the integral, we first must rewrite it as the following:

Now, perform the following subsitution:

Next, rewrite the integral and integrate:

The integral was performed using the following rule:

Finally, replace  with the  term:

To evaluate the integral, we must first make the following substitution:

Now, rewrite the integral and integrate:

The integral was performed using the following rule:

Finally, replace u with the x term:

To integrate, we must first make the following substitution:

Now, rewrite the integral in terms of u, and integrate:

The integration was performed using the following rule:

Finally, replace u with our original term:

To evaluate the integral, we first must make the following substitution:

Now, rewrite the integral, and integrate:

We used the following rule for integration:

Finally, replace  with our original term:

To evaluate the integral, first we must make the following substitution:

The derivative was found using the following rule:

Now, rewrite the integral and integrate:

The integral was performed using the following rule:

Finally, replace  with the  containing term:

Note that we removed the absolute value sign because the output of a square root is always positive.