We first rewrite the function

To solve the indefinite integral, we set．

Deriving then gives the equation, or．Substituting in forandgives the integral

Finding the anti derivative of this function we get

and replacingyields 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:

现在,路径替换itute 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.

现在,路径替换itute 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."

现在,路径替换itute 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:．堵在u和各自的值v, we get:

．

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

With this information,．

The entire denominator of the integrand, excluding the radical, can be rewritten assimply by replacingwith．

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, everymust be turned back to．

You know from the beginning of the problem that．This can be solved in terms ofby 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 ofis by using．Using the fact thatand．This can be found by knowing thatwhereand．

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) Changeto．Differentiatingfrom part (1):

4) Rewrite the integral in terms of: