Calculus/Indefinite integral

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DefinitionEdit

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   .

ExercisesEdit

1. Evaluate  

 

2. Find the general antiderivative of the function  

 

Solutions

Indefinite integral identitiesEdit

Basic Properties of Indefinite IntegralsEdit

Constant Rule for indefinite integrals

If   is a constant then  

Sum/Difference Rule for indefinite integrals

 
 

Indefinite integrals of PolynomialsEdit

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 functionEdit

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 functionEdit

Since

 

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

 

Integral of Sine and CosineEdit

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,

   
 
  .

ExercisesEdit

3. Evaluate  

 

4. Evaluate  

 

Solutions

The Substitution RuleEdit

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 ExampleEdit

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

 

GeneralizationEdit

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 RuleEdit

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.

ExamplesEdit

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 perform the following substitution:

 
 

Which yields:

 
 
 
 
 
 
 

ExercisesEdit

5. Evaluate   by making the substitution  

 

6. Evaluate  

 

Solutions

Integration by PartsEdit

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 ExampleEdit

General Integration by PartsEdit

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

ExamplesEdit

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  

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

 
 

ExercisesEdit

7. Evaluate   using integration by parts with   and  

 

8. Evaluate  

 

Solutions

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Indefinite integral