Chapters

## Integration

Integration is an important concept of calculus. When you integrate an equation, you are simply finding the area beneath that equation’s graph. To understand better, take a linear equation:

We can easily find the area beneath this line by finding the distance between the line and the axis, or even by simply counting the squares underneath it. Take a look at another example:

While we can count the area with whole squares, how can we measure the other squares? Do we take only half, or a third? This is where integration comes in:

The best Maths tutors available
1st lesson free!
4.9 (26 reviews)
Intasar
£36
/h
1st lesson free!
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
4.9 (13 reviews)
Paolo
£25
/h
1st lesson free!
4.9 (7 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!
5 (28 reviews)
Ayush
£60
/h
1st lesson free!
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
5 (14 reviews)
Farooq
£40
/h
1st lesson free!
5 (9 reviews)
Tom
£22
/h
1st lesson free!
4.9 (26 reviews)
Intasar
£36
/h
1st lesson free!
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
4.9 (13 reviews)
Paolo
£25
/h
1st lesson free!
4.9 (7 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!
5 (28 reviews)
Ayush
£60
/h
1st lesson free!
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
5 (14 reviews)
Farooq
£40
/h
1st lesson free!
5 (9 reviews)
Tom
£22
/h

## Integration Notation

When we’re interested in integrating a function, this is also known as the reverse of the derivative.

 A B C Function First derivative of the function Integral of the function

Take a look at the notation of integration:

 A Integration symbol B Function C This tells you which variable to integrate

An integral can be either definite or indefinite.

 B A Definite Indefinite The area between a defined interval The area of the entire function

You will mostly encounter definite intervals.

## Constants

Constants are the easiest to integrate. The rule for integrating constants can be found in the table below.

 Integral Rule Example Constant ax + c = 4x + c

The reason why we put a constant c is because the derivative of any constant is zero.

 Function Derivative Integral 4x + 30 f’ = 4x + 0 4x dx = 4x + c 4x + 2 f’ = 4x + 0 4x dx = 4x + c 4x f’ = 4x 4x dx = 4x + c

Adding the constant c to the integral reflects the fact that there might have been a constant in the original function.

Let’s take the following as an example:

To find the area, we simply subtract the upper interval and the lower interval.

## Exponents

Integrating exponents follows a similar process as above. Take a look at the exponent rule below.

This rule should be applied to every exponent you want to integrate. Let’s take a look at an example.

The table below has the steps for integrating this exponent.

 Step 1 Follow the rule specified above Step 2 Simplify the fraction

## Fractions

To integrate fractions, you will need a mix of the exponent rule as well as the following rules.

 Power/Fraction Rule = Reciprocal Integral Rule = ln|x| Integration by Parts = U-substitution Integration =

Let’s take one example and integrate it two ways. First, let’s rewrite the function using the power/fraction rule.

 Step 1 Rewrite the function = Step 2 Integrate using the exponent rule Step 3 Simplify =

Next, we can use u-substitution and the exponent rules. First, we write down our parameters.

 u = du dx x x^{-2}\sqrt{x} = \sqrt{\frac{1}{u}}x = \sqrt{\frac{1}{u}} = (u^{-1})^{\frac{1}{2}}x = u^{\frac{1}{2}}

Next, we replace the function with our parameters above.

We also have to replace the x in our function.

Simplify the u terms.

Now, integrate the function.

## Exponential and Logarithms

In order to integrate exponential or logarithmic function, you will need to follow the rules for integration.

### Exponential Function

Recall that an exponential function is one that has the variable in the exponent. Take a look at the notation and an example below.

 Notation Example

In order to integrate this function, we need to follow the following rule.

 Integral Exponential Integral Rule Example

### Euler's number

Euler's has a fixed value, which is e approximately equal to 2.718. Take a look at the notation and an example below.

 Notation Example

In order to integrate this function, we need to follow the following rule.

 Integral Euler's Integral Rule Example =

### Natural Logarithm

Natural logarithms have Euler’s number as the base. Take a look at the notation and an example below.

 Notation Example

In order to integrate this function, we need to follow the following rule.

## Sum Rule

When you have to integrate the sum of two functions, you simply integrate each function separately.

 Integral Function Rule

## Difference Rule

When you have to integrate the difference of two functions, it is the same as when you have the sum of two functions.

 Integral Function Rule

Need a Maths teacher?

Did you like the article?

3.00/5 - 2 vote(s)