Calculus/Indefinite integral

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Indefinite integral

Definition[edit]

Now recall that is said to be an antiderivative of f if . However, is not the only antiderivative. We can add any constant to without changing the derivative. With this, we define the indefinite integral as follows:

where satisfies and is any constant.

The function , the function being integrated, is known as the integrand. Note that the indefinite integral yields a family of functions.

Example

Since the derivative of is , the general antiderivative of is plus a constant. Thus,

Example: Finding antiderivatives

Let's take a look at . How would we go about finding the integral of this function? Recall the rule from differentiation that

In our circumstance, we have:

This is a start! We now know that the function we seek will have a power of 3 in it. How would we get the constant of 6? Well,

Thus, we say that is an antiderivative of .

Exercises[edit]

1. Evaluate

2. Find the general antiderivative of the function

Solutions

Indefinite integral identities[edit]

Basic Properties of Indefinite Integrals[edit]

Constant Rule for indefinite integrals

If is a constant then

Sum/Difference Rule for indefinite integrals

Indefinite integrals of Polynomials[edit]

Say we are given a function of the form, , and would like to determine the antiderivative of . Considering that

we have the following rule for indefinite integrals:

Power rule for indefinite integrals

for all

Integral of the Inverse function[edit]

To integrate , we should first remember

Therefore, since is the derivative of we can conclude that

Note that the polynomial integration rule does not apply when the exponent is . This technique of integration must be used instead. Since the argument of the natural logarithm function must be positive (on the real line), the absolute value signs are added around its argument to ensure that the argument is positive.

Integral of the Exponential function[edit]

Since

we see that is its own antiderivative. This allows us to find the integral of an exponential function:

Integral of Sine and Cosine[edit]

Recall that

So is an antiderivative of and is an antiderivative of . Hence we get the following rules for integrating and


We will find how to integrate more complicated trigonometric functions in the chapter on integration techniques.

Example

Suppose we want to integrate the function . An application of the sum rule from above allows us to use the power rule and our rule for integrating as follows,

.

Exercises[edit]

3. Evaluate

4. Evaluate

Solutions

The Substitution Rule[edit]

The substitution rule is a valuable asset in the toolbox of any integration greasemonkey. It is essentially the chain rule (a differentiation technique you should be familiar with) in reverse. First, let's take a look at an example:

Preliminary Example[edit]

Suppose we want to find . That is, we want to find a function such that its derivative equals . Stated yet another way, we want to find an antiderivative of . Since differentiates to , as a first guess we might try the function . But by the Chain Rule,

Which is almost what we want apart from the fact that there is an extra factor of 2 in front. But this is easily dealt with because we can divide by a constant (in this case 2). So,

Thus, we have discovered a function, , whose derivative is . That is, is an antiderivative of . This gives us

Generalization[edit]

In fact, this technique will work for more general integrands. Suppose is a differentiable function. Then to evaluate we just have to notice that by the Chain Rule

As long as is continuous we have that

Now the right hand side of this equation is just the integral of but with respect to . If we write instead of this becomes

So, for instance, if we have worked out that

General Substitution Rule[edit]

Now there was nothing special about using the cosine function in the discussion above, and it could be replaced by any other function. Doing this gives us the substitution rule for indefinite integrals:

Substitution rule for indefinite integrals
Assume is differentiable with continuous derivative and that is continuous on the range of . Then

Notice that it looks like you can "cancel" in the expression to leave just a . This does not really make any sense because is not a fraction. But it's a good way to remember the substitution rule.

Examples[edit]

The following example shows how powerful a technique substitution can be. At first glance the following integral seems intractable, but after a little simplification, it's possible to tackle using substitution.

Example

We will show that

First, we re-write the integral:

Now we preform the following substitution:

Which yields:

Exercises[edit]

5. Evaluate by making the substitution

6. Evaluate

Solutions

Integration by Parts[edit]

Integration by parts is another powerful tool for integration. It was mentioned above that one could consider integration by substitution as an application of the chain rule in reverse. In a similar manner, one may consider integration by parts as the product rule in reverse.

Preliminary Example[edit]

General Integration by Parts[edit]

Integration by parts for indefinite integrals
Suppose and are differentiable and their derivatives are continuous. Then

it is also very important to notice that

is not equal to

to set the and we need to follow the rule called I.L.A.T.E.


ILATE defines the order in which we must set the

  • I for inverse trigonometric function
  • L for log functions
  • A for algebraic functions
  • T for trigonometric functions
  • E for exponential function


f(x) and g(x) must be in the order of ILATE or else your final answers will not match with the main key

Examples[edit]

Example

Find

Here we let:

, so that ,
, so that .

Then:

Example

Find

In this example we will have to use integration by parts twice.

Here we let

, so that ,
, so that .

Then:

Now to calculate the last integral we use integration by parts again. Let

, so that ,
, so that

and integrating by parts gives

So, finally we obtain

Example

Find

The trick here is to write this integral as

Now let

so ,
so .

Then using integration by parts,

Example

Find

Again the trick here is to write the integrand as . Then let

so
so

so using integration by parts,

Example

Find

This example uses integration by parts twice. First let,

so
so

so

Now, to evaluate the remaining integral, we use integration by parts again, with

so
so </math>dv=\sin(x)dx</math>

Then

Putting these together, we have

Notice that the same integral shows up on both sides of this equation, but with opposite signs. The integral does not cancel; it doubles when we add the integral to both sides to get

Exercises[edit]

7. Evaluate using integration by parts with and

8. Evaluate

Solutions

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Indefinite integral