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 :

Solve via integration by parts. Make the following substitutions:

    .

Plug in substitutions: .

Solve via integration by parts again. Make the following substitutions:

      .

Plug in substitutions: .

Simplify: .

Plug this integral back into our original equation:

Simplify: .

Add  to both sides of the equation: .

Factor  out of the right side of the equation: 

Divide both sides of the equation by : 

Factor out a sinx:

Apply Pythagorean identity to : .

Making the following substitutions:   

Apply substitutions: 

Solve integral: 

Convert u back to x: 

Make the following substitution:  .

Plug the substitution into the integrand: .

Factor out  from the denominator, and simplify: .

Apply Pythagorean identity to the denominator, and simplify:.

Solve integral:.

Use original substitution to solve for :        . 

Plug value for  back into solution for integral: .

This integral, , is a classic integration by substation problem. This is indicated by the presence of a composite function in which both a function and its derivative are present. 

For this we need to let our variable  , substituting this into our integral we produce the following:

and by substituting back in for u we find our final answer to be:
.

                                                    (1)

To compute this integral we will use a trigonometric substitution. First we need to do some algebra to write the integrated in a more suitable form, 

                                               (2)

Now we can apply a trigonometric substitution. 

                                                   (3)

                                         (4)

Notice we've chosen this substitution so that the radicand in equation (2) will conveniently reduce as follows, 

So now we have  , we need to convert back to , use equation (3),