In order to integrate, we must use integration by parts, which follows the formula

We must set our  to be a function easy to differentiate:

This makes the integration step very easy later on!

The derivative was found using the following rule:

The rest of our given integrand is :

The integral was found using the following rule:

(We substitute

into the integral and integrate according to the above rule. Finally we replace  with our  term from above.)

Now, write out the integration by parts:

Finally, integrate:

The integration was performed using the following rule:

Our final answer is henceforth

To evaluate the integral, we must sum two integrals:

The first integral is equal to 

and was found using the following rule:

The second integral is found by first making a subsitution:

Now, rewrite the integral and solve:

We used the following rule to integrate:

Finally, replace  with our original  term and add the results of the integrations together:

To integrate, we must rewrite the integral into the sum of two integrals:

The first integral is equal to

and was found using the rule identical to the integral itself.

The second integral must be solved using integration by parts, which is given by

We are going to let

which means

The derivative and integral were found using the following rules:

, 

Now, use the above formula and integrate:

The integral was performed using the same rule as above.

Putting everything together, we get our final answer:

The integration of the function is easiest performed when you sum two separate integrals:

The first integral is equal to

and was found using the following rule:

The second integration is done by first making a substitution:

Now, rewrite the second integral and solve:

The integral was found using the following rule:

Now, replace u with the original term and add the two integrals together to get

This follows the basic intergral form for the Inverse Tangent. The form is

,where is a constant.

If you don't immediately see how matches the Inverse Tangent integral, then rewrite so that only the is in the denominator.

Then rewrite the as ,and the as , so they match the and the  in the integral form.

This matches the Inverse Tangent integral exactly with "u", "a", and "du" as follows

Now from the basic integral form, we can simply plug in "u" and "a".

This gives the correct answer of

To evaluate the integral, we must integrate by parts:

So, we must assign  and :

Next, we derive and integrate:

The derivative and integral were performed using the following rules:

, 

Next, use the above formula and integrate:

The following rule was used to integrate:

In order to calculate the indefinite integral, we apply the trigonometric property which states

As such, the indefinite integral is

In order to calculate the indefinite integral, we apply the inverse power rule which states

Applying this to the problem in this question we get

As such the indefinite integral is

The indefinite integral can be split into two separate integrals.

Evaluate each integral.

The answer to the first integral is:  

Be careful since the answer to  is not  as this is the derivative of .  In order to solve this integral,  we will need to use integration by parts.

If we let , we will have  by differentiation, and if we let , we will have  by integration.  The constant can be added in at the end of the problem.

Write the formula for integration by parts.

Substitute the terms into the formula.

Simplify the terms inside the integral and evaluate.

The answer to the second integral is:  

Combine the two answers.  The constants  can be combined to be a single constant term  at the end.

The answer is:  

This integral will require substitution with both  and  terms.

If we let , then  .

Differentiate  with respect to .

Substitute all terms back into the integral.

The  term is also the same as , which can be multiplied within the terms of the parentheses.  Simplify the integral.

Evaluate this integral.

Resubstitute .

The answer is: