# 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.

$\int&space;(f(x)&space;\pm&space;g(x))dx=\int&space;f(x)dx&space;\pm&space;\int&space;g(x)dx$

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

$\int&space;f(x)\pm&space;g(x)dx$

makes no sense.

Also,

$\int&space;1&space;dx=\int&space;dx&space;=&space;x+C$

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.

$\int&space;f(t)&space;dx=f(t)\int&space;dx&space;=&space;f(t)x+C$

Example:

$\int&space;5k&space;dx=5k\int&space;dx&space;=&space;5kx+C$

Constant rule:

Constants are taken out of the integration sign.

$\int&space;g(t)f(x)&space;dx=g(t)\int&space;f(x)dx$

Example:

$\int&space;k^2\ln(x)&space;dx=k^2\int&space;\ln(x)dx$

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

Simmetry

$\int_{-a}^a&space;\left&space;\{&space;\text{even&space;function}\right&space;\}dx=2\int_0^a\left&space;\{&space;\text{even&space;function}\right&space;\}dx$

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:

$\int_{-2}^2&space;(\left&space;|&space;3x&space;\right&space;|-7x^4)dx=2\int_0^2(\left&space;|&space;3x&space;\right&space;|-7x^4)dx$
$\int_{-a}^a&space;\left&space;\{&space;\text{odd&space;function}\right&space;\}dx=0$

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

$\int_a^bf(x)dx+&space;\int_b^cf(x)dx=\int_a^cf(x)dx$

Demonstration:

$\int_a^bf(x)dx+&space;\int_b^cf(x)dx=\left&space;[&space;F(b)-F(a)&space;\right&space;]+&space;\left&space;[&space;F(c)-F(b)&space;\right&space;]=$ $-F(a)&space;+&space;F(c)&space;=&space;F(c)-F(a)$ $\text{in&space;fact,&space;}\int_a^cf(x)dx$

Example:

$\int_{-\frac{\pi}{4}}^\frac{\pi}{2}\cos(x)dx+\int_\frac{\pi}{2}^\pi\cos(x)dx=\int_{-\frac{\pi}{4}}^\pi\cos(x)dx=$ $\left&space;[&space;\sin(x)&space;\right&space;]_{-\frac{\pi}{4}}^\pi=\sin(\pi)-\sin\left&space;(&space;-\frac{\pi}{4}&space;\right&space;)=0-\left&space;(&space;-\frac{\sqrt2}{2}&space;\right&space;)=\frac{\sqrt2}{2}$

The following is a little introduction to integration techniques.

$\int&space;f(x)g(x)dx=f(x)\int&space;g(x)dx-\int&space;\left&space;(&space;\frac{\mathrm{d}&space;}{\mathrm{d}&space;x}f(x)\int&space;g(x)dx&space;\right&space;)dx=$ $f(x)G(x)-\int&space;f'(x)G(x)dx$

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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