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:

integration_definite_example

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:

integration_what_is_it

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:

integration_parabola

 

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Integration Notation

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

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

 

Take a look at the notation of integration:

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

 

An integral can be either definite or indefinite.

definite_indefinite_integration

 

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 \int a dx ax + c \int 4 dx = 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 \int 4x dx = 4x + c
4x + 2 f’ = 4x + 0 \int 4x dx = 4x + c
4x f’ = 4x \int 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:

definite_integration_example

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

definite_integration_step_by_step

 

Exponents

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

integration_exponent_rule

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

exponent_integration_example

The table below has the steps for integrating this exponent.

 

Step 1 Follow the rule specified above \int x^{3} dx = \frac{x^{3+1}}{3+1}
Step 2 Simplify the fraction \frac{x^{4}}{4}

 

Fractions

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

 

Power/Fraction Rule \frac{k}{x^{n}} = k*x^{-n}
Reciprocal Integral Rule \int \frac{1}{x} dx = ln|x|
Integration by Parts \int f(x)*g(x) dx = f(x)*(\int g(x) dx) - \int f(x) * (\ing g(x) dx) dx
U-substitution Integration \int f(g(x)) * g'(x) dx = \int f(u) du

 

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

fraction_integration

 

Step 1 Rewrite the function \int \frac{1}{x^{2}} dx = \int x^{-2} dx
Step 2 Integrate using the exponent rule \frac{x^{-2+1}}{-2 + 1}
Step 3 Simplify \frac{x^{-1}}{-1} = - frac{1}{x}

 

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

 

u \frac{1}{x^{2}} = x^{-2}
du -2x^{-3}dx
dx du = \frac{-2}{x^{3}} dx

dx = - \frac{1}{2}x^{3} du

x u =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.

u_substitution_rules

We also have to replace the x in our function.

u_substitution_example

Simplify the u terms.

integration_u_substitution_example

Now, integrate the function.

u_substitution_fraction

 

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
f(x) = a^{x} f(x) = a^{5}

 

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

 

Integral Exponential Integral Rule Example
\int a^{x} dx \frac{a^{x}}{ln(a)} \int a^{5} dx = \frac{a^{5}}{ln(5)}

 

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
f(x) = e^{x} f(x) = e^{3}

 

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

 

Integral Euler's Integral Rule Example
\int e^{x} dx e^{x} \int e^{3} dx

= e^{3}

 

Natural Logarithm

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

 

Notation Example
f(x) = ln(x) f(x) = ln(3)

 

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

 

Integral Natural Log Integral Rule Example
\int ln(x) dx x(ln(x)) - x \int ln(2) dx = 2(ln(x)) - 2

 

Sum Rule

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

 

Integral Function Rule
\int (f(x) + g(x)) dx \int f(x) dx + \int g(x) dx

 

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
\int (f(x) - g(x)) dx \int f(x) dx - \int g(x) dx

 

 
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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.