Rules and properties of integrals

Ok, this is the second step! I will cover the rules you need to apply to calculate an integral. With the next lesson, you will hopefully be able to start integrating! Let’s start! But first, I recommend you check out my post “What’s integration?” https://addjustabitofpi.com/2019/09/28/whats-integration/

Linearity

The sum or difference of two (or more) functions under the integral sign is equal to the sum or difference of the integral of each function.

Remember to use the parenthesis if you want to integrate a sum or a difference, because

makes no sense.

Also,

and more in general, whatever is not a function of the variable of integration, here x, is a constant and as such is taken out of the integral sign.

Example:


Constant rule:

Constants are taken out of the integration sign.

Example:


These rules are valid for definite integrals as well, but the latter have some other rules:

Simmetry

Examples of even function are x², cos(x), abs(x). It means that, whether the x is negative of positive, the result is going to be the same. In fact, 5²=(-5)²=25, cos(60) = cos(-60) = 0.5, abs(12) = abs(-12)=12 and so on.

Example:


Examples of odd functions are x^3, sin(x), x. It means that 2³ != (-2)³, in fact 8 != -8; sin(30) != sin(-30) (0.5 and -0.5) and so on.


Continuous bounds

Demonstration:

Example:


The following is a little introduction to integration techniques.

but I will talk about it in a more detailed way in the next post.


Once you memorise these rules and properties, it will be easier to understand how to actually calculate integrals. Wait for it!

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