Fundamental Theorem of Calculus and Techniques of Antidifferentiation - AP Calculus BC

First, we will find the indefinite integral, .

We will let and .

Then,

and .

and

Now, this expression evaluated at is equal to

.

At it is undefined, because does not exist.

We can use L'Hospital's rule to find its limit as , as follows:

and , so by L'Hospital's rule,

Therefore,

Rewrite the integral as

.

Substitute . Then

and . The lower bound of integration stays , and the upper bound becomes , so

Since , the above is equal to

.

By the Formula Rule, we know that . We therefore know that .

Continuing the calculation:

By the Power Rule for Integrals, for all with an arbitrary constant of integration . Therefore:

.

So,

.

In order to evaluate this integral, we will need to use partial fraction decomposition.

Multiply both sides of the equation by the common denominator, which is

This means that must equal 1, and

The answer is .

To integrate, we must first make the following substitution:

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

The following rule was used to integrate:

Finally, we replace u with our original x term:

By the Fundamental Theorem of Calculus, we have that . Thus, .

Set . Then , and by the chain rule,

By the fundamental theorem of Calculus, the above can be rewritten as

We can view the function as a function of , as so

where .

We can find the derivative of using the chain rule:

where can be found using the fundamental theorem of calculus:

So we get

Via the Fundamental Theorem of Calculus, we know that, given a function, .

Therefore .

Via the Fundamental Theorem of Calculus, we know that, given a function , . Therefore, .

By the Fundamental Theorem of Calculus, for all functions that are continuously defined on the interval with in and for all functions defined by by , we know that .

Thus, for

,

.

Therefore,

By the Fundamental Theorem of Calculus, for all functions that are continuously defined on the interval with in and for all functions defined by by , we know that .

Given

, then

.

Therefore,

.

Use the fundamental theorem of calculus to evaluate:

Use the Fundamental Theorem of Calculus and evaluate the integral at both endpoints:

Use the Fundamental Theorem of Calculus and evaluate the integral at both endpoints:

Evaluate the following integral

Let's begin by recalling our "reverse power rule" AKA, the antiderivative form of our power rule.

In other words, all we need to do for each term is increase the exponent by 1 and then divide by that number.

Let's clean it up a little to get:

Now, to evaluate our integral, we need to plug in 5 and 0 for x and find the difference between the values. In other words, if our integrated function is F(x), we need to find F(5)-F(0).

Let's start with F(5)

Next, let's look at F(0). If you look at our function carefully, you will notice that F(0) will cancel out all of our terms except for +c. So, we have the following:

Finding the difference cancels out the c's and leaves us with 185.

In order to evaluate this integral, we will need to use partial fraction decomposition.

Multiply both sides of the equation by the common denominator, which is

This means that must equal 1, and

The answer is .

To integrate, we must first make the following substitution:

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

The following rule was used to integrate:

Finally, we replace u with our original x term:

First, we will find the indefinite integral using integration by parts.

We will let and .

Then and .

To find , we use another integration by parts:

, which means that , and

, which means that, again, .

Since

, or,

for all real , and

,

by the Squeeze Theorem,

.