Here, we can use u-substitution. We'll set  and we'll factor out the  outside the integral.

Now let's calculate 

And solve for dx.

.

Plugging these values into the integral we now get

.

We now see that the 's cancel out and we're left with an integral entirely with u.

.

We just need to replace u with its original value, doing so results in our final solution.

To solve the following integral, we must make a substitution to create the following general form:

We make the following subsitution:

The derivative was found using the following rule:

The integral now looks like this:

Notice that it is in the same form as the integral we want.

Now use the form from above to integrate:

To finish the problem, replace u with 2x:

.

We can simplify

by first doing a substitution, with , which gives us , which means that . So the integral becomes

The integral can be solved using two integration by parts, which will give us 

So now we just plg in  into  and  into  to get

Pull out the constant out in front of the integral.

Use U-substitution to solve.

The integral can be solved with a clever substitution:

The derivative was found using the following rules:

,

Then, when you rewrite the integral in terms of u, you find that you get:

The integration was performed using the following rule:

Finally, replace the u with our original term.

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 

.