Integration by Recognition
Neil Trivedi
Teacher
Contents
Introduction to Integration by Recognition
Integration is the inverse process of differentiation. In other words, if differentiating a function gives , then integrating gives (with a constant of integration added for indefinite integrals).
So, when we integrate, we are asking ourselves “what function differentiates to give this?”
Integration by recognition often involves “guesstimation,” where we make an educated guess for and then check it by differentiating. If the derivative of the guess differs from our function by a constant factor , we simply adjust the guess by multiplying by to get the correct integral. We should never be dividing by variables, only constants.
Note: “Guesstimation” is a term totally made up by NeilDoesMaths. We are never actually “guessing.” Instead, we are recognising the form of the integral which comes naturally through experience and exposure to various integrals. In our working out, we will use the “guess” to represent our recognised integral.
General Steps when Integrating by Recognition
1) Make a guess (or guesses if there’s more than one term) as to what may differentiate to give us the function we’re integrating.
2) Differentiate the guess.
3) If the derivative is a constant multiple of our function, adjust by multiplying the guess by .
4) Don’t forget to add a at the end (for indefinite integrals).
Power Functions
Example 1:
Integrate with respect to :
a)
Step 1: Make a guess as to what may differentiate to give .
We will guess . With power functions, the power goes up by .
Step 2: Differentiate the guess.
Step 3: Adjust the constants.
Our guess differentiates to give , but we wanted . So, we multiply both sides by to get
b)
Step 1: Make a guess as to what may differentiate to give .
We will guess . For power functions, we add to the power. To note, the in the question is what we call a “multiplier.” Always ignore multipliers when guessing and we will adjust the constants later.
Step 2: Differentiate the guess.
Step 3: Adjust the constants.
Our guess differentiates to give , but we wanted . So, we multiply both sides by to get
No answer provided.
Note: If you need another trick on how to adjust the constants, just remember that you are multiplying both sides by what you “want” divided by what you “have”. So, multiply by the constant we want over the constant we have.
Trigonometric Functions
Example 2:
Integrate with respect to :
a)
Step 1: Make a guess as to what may differentiate to give .
We will guess . Like with differentiating trigonometric functions, always keep the angle the same.
Step 2: Differentiate the guess.
Step 3: Adjust the constants.
Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which will be .
b)
Step 1: Make a guess as to what may differentiate to give .
Since we know differentiates to the form , we will guess . Remember, keep the angle the same.
Step 2: Differentiate the guess.
Step 3: Adjust the constants.
Our guess differentiates to give , but is what we wanted. So, we multiply both sides by the constant we want over the constant we have, which will be .
No answer provided.
Fractions
Example 3:
Integrate with respect to :
a)
Step 1: Make a guess as to what may differentiate to give .
We will guess . The is a multiplier so we can ignore that in our guess.
Step 2: Differentiate the guess.
Step 3: Adjust the constants.
Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which will be .
b)
Step 1: Make a guess as to what may differentiate to give .
We will guess . The is a multiplier so we can ignore that in our guess.
Step 2: Differentiate the guess.
Step 3: Adjust the constants.
Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which will be .
No answer provided.
Exponentials
Example 4:
Integrate with respect to :
a)
Step 1: Make a guess as to what may differentiate to give .
For all exponentials, they generally differentiate to the same function (with a constant that we will adjust for later) and so, when integrating, we always guess the same function. Hence, we will guess . The is a multiplier so we can ignore that in our guess.
Step 2: Differentiate the guess.
Step 3: Adjust the constants.
Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which will be .
b)
Step 1: Make a guess as to what may differentiate to give .
For all exponentials, they generally differentiate to the same function (with a constant that we will adjust for later) and so, when integrating, we always guess the same function. Hence, we will guess .
Step 2: Differentiate the guess.
Step 3: Adjust the constants.
Our guess differentiates to give , but we wanted . So,
we multiply both sides by the constant we want over the constant we have, which will be
.
Note: Here, it looks cleaner to write all as one fraction, but you can also write the answer as if you want.
No answer provided.
Mixed Functions
Example 5:
Integrate with respect to :
a)
Note: We will make two separate guesses and integrate each term individually.
Step 1: Make guesses as to what may differentiate to give each term.
For , we will guess (remember to ignore the as it is a multiplier) and for , we will guess .
Step 2: Differentiate the guesses.
Step 3: Adjust the constants.
• Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which will be .
• Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which will be .
b)
Note: Just like part (a), we will make two separate guesses and integrate each term individually.
We may also rewrite as .
Step 1: Make guesses as to what may differentiate to give each term.
For , we will guess ( is a multiplier so we can ignore it) and for , we will guess (for power functions, we add to the power.)
Step 2: Differentiate the guesses.
Step 3: Adjust the constants.
• Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which will be .
• Our guess differentiates to give , but we wanted . So, we
multiply both sides by the constant we want over the constant we have, which will be
.
No answer provided.
Practice Questions