Rules and properties of integrals

3 Oct 2019 1 Comment

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://atomic-temporary-167386734.wpcomstaging.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.

$\int (f(x) \pm g(x))dx=\int f(x)dx \pm \int g(x)dx$

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

$\int f(x)\pm g(x)dx$

makes no sense.

Also,

$\int 1 dx=\int dx = x+C$

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.

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

Example:

$\int 5k dx=5k\int dx = 5kx+C$

Constant rule:

Constants are taken out of the integration sign.

$\int g(t)f(x) dx=g(t)\int f(x)dx$

Example:

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

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

Simmetry

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

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:

$\int_{-2}^2 (\left | 3x \right |-7x^4)dx=2\int_0^2(\left | 3x \right |-7x^4)dx$

$\int_{-a}^a \left \{ \text{odd function}\right \}dx=0$

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

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

Demonstration:

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

Example:

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

The following is a little introduction to integration techniques.

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

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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Integration Calculus, Indefinite integrals, properties of integrals, rules of integration

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Rules and properties of integrals

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