We substitute 

to solve the integral. Solving for dx,

Substituting these values into the integral yields

Solving for  from

gives us

And so the indefinite integral is

The integrand is composed of a function as well as its derivative multiplied by a constant. Hence, we can find the antiderivative via u-substitution as follows:

Let . Then , and so . Thus,

The integrand can be evaluated by means of the u-substitution method, as follows:

Let . Then , and so

Here, an understanding of trigonometric identities, as well as the appropriate selection of a dummy variable for u-substitution, is required. To figure out which function to represent "u" (cosine or sine), simply re-write the integrand as

Remembering that ,

Now, we can substitute  to yield

because if , then , which implies .

At this point, all that is left to do is expand the polynomial and evaluate the integrand:

To perform this integration, we use a substitution.

Since the derivative of is , we choose our substitution to be .

Differentiating gives us,

 .

Now we can substitute this into our integral. We will have,

 .

Along with this substitution, we must also change our limits of and . To do so, we take these values and plug them in for  in the formula .

Doing so, we obtain and .

Now our integral will be transformed as follows,

 .

This integral is now easy to integrate, for the function integrates to .

Thus we have,

 .

Therefore, the answer to the integral is,

.

To integrate, we must first make the following substitution:

Now, rewrite the integral in terms of u and integrate:

The integral was performed using the following rule:

Note that the rule contains a fraction in front of the inverse trig function. Do not confuse this fraction with the fraction coming from the u substitution!

Finally, replace u with our original term and multiply the constants:

To evaluate the integral, we can first use the fact that cosine and secant are inverses of each other, so they cancel:

Now, we must make the following substitution:

Rewriting the integral in terms of u and integrating, we get

We used the following rule to integrate:

Finally, replace u with our original x term:

To integrate this problem, you have to use "u" substitution. Assign . Then, find du, which is 2x. That works out since we can then replace the other x in the original problem. We will have to offset the 2 though: . Now plug in all the parts: . Now, integrate as normal, remembering to raise the exponent by 1 and then also putting that result on the bottom: . Simplify, add a C because it is an indefinite integral, and substitute your original expression back in: .

To integrate this problem, use "u" substitution. Assign , . Substitute everything in so you can integrate: . Recall that when there is a single variable on the denominator, the integral is ln of that term. Therefore, after integrating, you get . Sub back in your original expression and add C because it is an indefinite integral: .

Make the substitution: