In this case, a u-substitution can be used to evaluate this integral. Specifically:

, , 

Next, the intergand must be rewritten in terms of u:

The integral was found by using the following rule:

Finally, u must be replaced with x, giving a final answer of:

The only apparent way to evaluate this integral is by using a trig substitution. The denominator resembles the form  which means that . Specifically:

, 

With a value of x found, the radical portion of the denominator can be rewritten:

The final equation was solved by using the following trigonometric identity:

Now, every x must be replaced with theta in the integrand:

There is not an easy way to evaluate this integral, but it can be rewritten as:

Even though an integral was evaluated, it must be expressed in terms of x. To do this, refer back to the equation, . This can be rearranged to:

With this piece of information, a right triangle can be created with hypotenuse equal to seven and opposite side equal to x. After using the Pythagorean Theorem, , to solve for the rest of the triangle, you should find that:

Therefore, the final answer is:

The only way to solve evaluate the integral is by splitting it into two separate, solvable integrals. Specifically, the integrand must be rewritten as:

The first integral must be rewritten as:

For the second integral, both the numerator and the denominator must be rewritten:

After combining the sums of each integral, the final answer is:

A u-substitution will work for evaluating the integral. Specifically:

, , 

Now, the integrand must be expressed in terms of u in order to evaluate the integral:

The integral was finally taken by using the following rule:

After replacing u with x, the final answer is:

The only way to evaluate this integral is by first expanding the integrand:

Next, the integrand must be evaluated as three simpler integrals:

The first integrand must be rewritten as:

The second integrand must be rewritten as:

Finally, the third integral simply equals x+C because of the following rule

After adding all of the smaller integrals together, the final answer is:

The only way to evaluate this integral is by doing a trig substitution. The radical portion of the denominator resembles the form  which means that . Specifically:

, 

With a known value of x, the radical portion of the denominator can be rewritten as:

The final portion of the equation came from the following trigonometric identity:

Next, the integrand must be written entirely in terms of theta:

After simplifying, you are left with:

There is no way to evaluate this integral other than rewriting the integrand using the half-angle identity for cosine:

The new integrand can be split and evaluated as two separate integrals:

Even though an answer has been found, it must be in terms of x. Refer back to the equation, . This can be rearranged to . From this, you can find that . Additionally, by using the Pythagorean Theorem, , you can find any trigonometric function. The double-angle sine must be rewritten using the following double-angle idenetity:

Specifically,  and 

After rewriting theta in terms of x, the final answer is:

The only way to evaluate this integral is through integration by parts. To do this, you must follow the equation:

You must assign values for u and dv from the original integrand and then find the values of du and v. Specifically:

, , , 

From here, plug in the values into the equation for integrating by parts:

To evaluate this integral, a u-substitution is needed:

, 

Now, the integrand from the equation can be rewritten as:

When you replace u with x and add the other half of the equation, the final answer is:

The only way to evaluate this integral is through integration by parts where the formula states:

Values for u and dv must be picked from the integrand and then the remaining values are found from those. Specifically:

, , , 

Now, these values must be plugged into the equation:

The integral that now must be evaluated can be done so using a u-substitution:

, 

Next, every x must be replaced with u and integrated:

After replacing u with x and adding the remainder of the equation, the final answer is:

The only way to evaluate this integral is by splitting it in to two separate integrals:

The first integral can be evaluated by using a u-substitution:

, 

Then, x must be replaced with u and evaluated:

For the second integral, the integrand resembles the form:

In this case,  and the integral of it is 

After adding all of the evaluated integrals, the final answer is:

The only way to evaluate this integral is by recognizing that the integral resembles the following:

To evaluate the integral, the x-term must be replaced with u:

, , 

Now that the denominator is in the proper form, the integral can be evaluated according to the first equation:

U must be replaced with x to give the final answer: