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When looking at the problem, one would expect to rewrite the integrand using a u-substitution and it would make it more easily solvable. However, a u-substitution will not work in this instance, but a trigonometric substitution will work very well.

We can make the following substitution:

The derivative was found using the following rule:

Next, we can rewrite the integrand:

which simplified becomes

We can rewrite the integrand again as

The integral was found using the following rules:

, 

Now, our answer is in terms of theta, which we must change back in terms of x. To do this, we must use what we designated in the beginning as x. We draw a triangle, where the ratio of the hypotenuse to the side adjacent to angle theta is . Using the Pythagorean theorem to find the length of the third side of the triangle, and plugging in tangent and the value of theta itself, we get

There is not an easy way to integrate the function because a u-substitution would not work nor can it be simplified. However, any trinomial can be rewritten by completing the square.

In order to do this, the coefficient of the highest power term must be equal to one. In this case, it equals -1, so one must pull out a -1 from the entire function:

Once the function is in a form where completing the square is possible, separate the constant term from the rest of the function:

Then, divide the coefficient of the x-term by two and then square it. This number will be added to the new trinomial formed as well as to the twelve that was separated from the function.

Finally, the perfect square trinomial formed can be rewritten:

Now, the integrand is rewritten and can be integrated:

However, a u-substitution will not work in this instance, but a trigonometric substitution will work very well. The function takes after the form of the square root of a squared minus x squared.

We can make the following substitution:

The derivative was found using the following rules:

, 

Rewriting using Pythagorean identities, we get

There is not an easy way to integrate this. However, using the half-angle identity for , the integrand can be rewritten as

Integrating, we get

using the following integration rules:

, 

However, one must convert the thetas back to x’s in order to solve the original integral.

To do this, we must consider what we designated as x, which is 

Now, we must find theta:

Now, one can use the Pythagorean theorem to determine all of the sides of the triangle.

However, one must solve for  by using the double-angle identity, which is .

 equals  This is found by dividing the adjacent side of the triangle by the hypotenuse. Theta can be found by multiplying both sides of by  to get

Our final answer is

The problem looks intimidating, but we can simplify the integrand using the properties of logarithms to

Now, we can rewrite the integrand again using the double angle identity for sine:

Next, we can integrate using a u-substitution:

, 

Rewriting and integrating, we get

The integral was performed using the following rule:

Finally, replace u with our original x term:

To integrate, we must first make the following substitution:

, 

The derivative was found using the following rules:

, 

The radical is acting as the "outer" function using the first rule (the chain rule).

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

The integration was performed using the following rule:

Finally, replace u with our original x term:

Note that we were able to remove the absolute value sign because the square root will always be positive. 

Substitute , so .

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Replace u with : 

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For 

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first substitute 

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Replace u with :

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