We first rewrite the function

To solve the indefinite integral, we set .

Deriving then gives the equation , or . Substituting in for  and  gives the integral

Finding the anti derivative of this function we get

and replacing  yields the answer

To integrate, we must first make a substitution:

The derivative was found using the rule

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

The integral was found using the following rule:

Finally, replace u with our original x term:

First, assign u substitution in order to integrate the expression:

Now, substitute everything in so you can integrate:

Now, integrate. Remember when there is a single x on the denominator, the integral is ln of that term.

Now, substitute back in the initial expression and add a +C because it is an indefinite integral:

To integrate this expression, you'll have to use u substitution. Assign your "u."

Now, substitute everything in:

Integrate:

Substitute your original expression back in and add a C because it is an indefinite integral:

To solve this integral, we use u substitution. However, to do so, we must break our integral into two separate integrals, which looks like this:

Now that we have two separate integrals, we can make the appropriate substitutions for each one. For the first integral, we make the following substitution:     . For the second integral, we make this substitution:   . This changes our integral to: , which equals:. Plugging back in our respective values for u and v, we get:

 .

There are no apparent substitutions to rewrite the integrand with other than a trigonometric substitution. The denominator resembles  which means that . Specifically,  which means that .

With this information, . 

The entire denominator of the integrand, excluding the radical, can be rewritten as  simply by replacing  with . 

This can be simplified to . This comes from the trigonometric identity .

Now, this problem can be rewritten entirely in terms of :

The integral further simplifies to:

There is no way to evaluate this integral other than rewriting the integrand as:

This comes from the trigonometric identity:

Now, the integral can be easily evaluated by splitting the integrand:

The second integral was evaluated using the following:

The integral may seem to be evaluated. However, the original integral was in terms of . Therefore, every  must be turned back to . 

You know from the beginning of the problem that . This can be solved in terms of  by dividing both sides of the equation by 2 and then by taking the inverse sine of both sides, leaving you with:

The only way that the second term can be rewritten in terms of  is by using . Using the fact that and . This can be found by knowing that  where  and . 

The second term can now be rewritten as:

This simplifies to:

The final answer is now:

A u-substitution would properly simplify the integrand, where 

Now, the problem can be rewritten entirely in terms of u:

The problem may seem finished, but the original integrand was expressed in terms of . Therefore, the final answer is, in fact:

There are no apparent substitutions for solving this integral, but the integral can be expressed as the sum of two separate integrals because this is a property of indefinite integrals.

The first integral can be solved with a simple u-substitution where . 

The integral can be rewritten as:

To finally solve this, there is no other way to do so other than knowing the following: 

Finally, the answer must be expressed in terms of :

The second integral is a bit more complicated. It can be noted that the second integral resembles the following:

Specifically, . The second integral can be rewritten as:

With each separate integral found, the answers can be added to equal the original integral:

To solve this, you can use a u-substitution where

.

Now, the integrand can be completely rewritten in terms of u:

Before trying to further solve this integral in a more complicated way, by remembering what u equals, the integrand can be rewritten as:

The integral was taken by using the following formula:

The original problem was in terms of x. Therefore, the final answer is:

The trig sub is used to redefine the integral in terms of a different variable, , to make its evaluation possible. 

For the given integral, following steps are important to redefine it using trigonometric substitution:

1) Find out, which trigonometric identity is best for given integral. In our case, the integrand is: . Therefore,   is an appropriate identity.

2) Redefine the bounds of integration:

3) Change  to . Differentiating  from part (1):

4) Rewrite the integral in terms of :