There are no apparent substitutions to simplify the integrand, but you can rewrite it as:

With this expression, it s clear that you must solve the integral by using partial fractions. To do this, two fractions must be created that add up to the integrand. Specifically, the partial fractions will be:

Next, the denominator must be cleared in order to solve for the unknowns. To do this, the entire equation must be multiplied by a common denomionator, . The resulting equation is:

Next, a system of equations must be created by equating the coefficients of the terms with the same power of x. The result is:

After solving, you find that 

With these values found, they can be plugged back into the original equation and integrated:

The final answer is:

The integrals were evaluated by using the following rule:

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: