To integrate, the following substitution was made:

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

The following rule was used for integration:

Finally, rewrite the final answer in terms of our original x term:

Re-write the integrand as follows:

Make the following substitution:

Apply the substitution to the integrand:

Evaluate the integral:

Re-substitute the value of u:

Solution:

To integrate, we can split the integral up (the property of linearity allows us to do this):

The first integral is equal to

and was found using the following rule:

The second integral can be solved after the following substitution is made:

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

The integral was solved using the identical rule.

Next, rewrite the answer in terms of x:

Finally, add this to the first result to get our final answer:

Note that all of the integration constants were combined to make a single constant.

We are going to use U-substitution

Looking at the original

Let

Then

And now we can use our anti derivative rules (don't forget your constant!)

Finally, substitute back in for u

Solve by u substitution

Make the following substitution:

Apply the substitution the integral:

Solve the integral:

Re-substitute the value for u:

Evaluating integrals requires knowledge of the basic integral forms. In this problem, there is a group raised to an exponent, which is the .

This arrangement typically follows the basic integral form , where is a variable expression, is some constant number and is the constant of integration, which stays unknown. In this case, we can use u-substitution to match this form.

The exponent is 5, so the in the basic integral form will be5.

Since the inside of the exponent group is , set .

Now we differentiate to find . Recall that the derivative of .

Notice that the parts of our match up with the integral we are evaluating, and there are no variables that aren't accounted for. This is clearer if we write out and simplify what our substitution says.

This is exactly what we are asked to integrate.

This means we can evaluate the integral by using basic integral form directly. Plugging in our and into the right side of , we get

This is equivalent to the answer , which is the correct answer.

There are a lot of pieces inside this integral. There are trigonometry functions, exponential functions, and a rational arrangement. Lots of possibilities.

This is where u-substitution is best. Try making u represent various parts and see if du gets all the other parts. After doing this enough times, you will see that we should make u be the exponent of the e.

Let's write the inner fraction as an x with a negative exponent

Now we differentiate and see what we get for du. This will require the chain rule. The outer structure is the , which requires trig functions integration rules, and the chained inner structure is a power rule arrangement

Recall these derivatives:

Applying these we get

Simplifying gives us

This perfectly matches all remaining parts of our integral. Lets rewrite everything using u and du

This matches the basic integral form,

Thus, when we integrate we get

Rewriting back in terms of x, we get

This is our answer.

To integrate, the following substitution must be made:

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

The following integration rule was used:

To finish, we replace u with our original x term:

To integrate, the following substitution was made:

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

The following rule was used to integrate:

Finally, rewrite the answer in terms of x:

Rewrite the integrand as follows:

.

The integral can be rewritten as

Now, use -substitution, setting . It follows that

The limits of integration can be rewritten as

The integral becomes

Integrate:

,

the correct response.