The integral is found by recognizing the following integral:

To solve the integral, make the following substitution:

, 

The derivative was found using the following rule:

The integral then becomes

Finish by replacing u with the original term containing x.

Substitution can be used to make this problem easier. Let u = sin(x). Then du = cos(x)dx. This allows us to rewrite and evaluate the original integral in terms of u as:

The final answer should be written in terms of the original variables, so substitute sin(x) for u. 

Note that we could have also chosen cos(x) as u, but the above substitution avoids introducing negatives. 

First, notice that . Use the substitution  to rewrite the integral as

.

Next, recall that , so 

. 

Lastly, substitute  in place of  to write the answer in terms of the original variables. 

To solve the indefinite integral, make a simple substitution:

The integral then becomes:

After integrating, we get

The following rule was used for the integration:

Finally, replace the u with the original term containing x. 

We know that the derivative of   is  .

Doing a substitution and setting

 and  allows us to rewrite the integral as 

 which can be rewritten as .

Integrating this gets you  plus a constant (which is stated in the original question that you can assume that we already have one). Substituting  back in gives us the final answer, which is 

.

We know that the derivative of  is .

So substituting

 allows us to have 

.

This allows us to rewrite the integral as 

 which, when integrated, gives us 

.

Substiting x back in gives us the answer, 

.

To evaluate the integral, we must recognize that what we were given looks very similar to the following integral:

To make our integral look like the one above, we must perform the following substiution:

Now, rewrite our integral:

It looks like the one above, so we can integrate now:

Finally, replace u with our original term:

To integrate, we must break the integral into three integrals:

The first integral is equal to 

and was found using the following rule:

The second integral is equal to

and was found using the following rule:

The final integral is found by performing the following substitution:

Now, rewrite and integrate:

The integral was found using the following rule:

Finally, rewrite the integral in terms of  by replacing  with the original term, and add all three integrals together to get a final answer of 

To integrate, we must make the following substitution:

The derivative was found using the following rule:

Now, rewrite the integral:

Notice that we changed

Next, distribute and integrate:

The integral was found using the following rule:

Finally, replace  with our original  term:

To solve the integral, we must make the following substitution:

Now, rewrite the integral and integrate:

The integral was found using the following rule:

Finally, replace  with our original  term: