Fundamental integrals

In order to be able to solve integrals, (whether they are definite of indefinite) you need to learn the fundamental integrals first, the same way you did for the derivatives.

I suggest you check this out, if you haven’t yet:

$\int&space;\sin(x)dx=-\cos(x)&space;+C$

Remember: never forget to add the “+C” next to the result of an indefinite integral: this is because the derivative of a constant is zero, so if you take the derivative of this result, yes, you get sin(x) but the function f(x) = -cos(x) + 4, for example, is different from the function f(x) = -cos(x) – 12. Taking the derivative of both functions still gives you sin(x), but they are actually different.

Now you know, we can move on!

$\int&space;\cos(x)dx=\sin(x)&space;+C$ $\int&space;\frac{1}{x}dx=\ln(x)&space;+C$ $\int&space;e^xdx=e^x&space;+C$ $\int&space;{x^n}dx=\frac{x^{n+1}}{n+1}&space;+C&space;\text{&space;with&space;}n\neq&space;-1\text{,&space;since&space;}x^{-1}=&space;\frac{1}{x}\text{&space;and&space;the&space;integral&space;is&space;}\ln(x)$

Believe it or not, these five integrals are enough to calculate every integral. For this though, we need to know the different techniques of integration, which I will cover in another post. Also, when calculating an integral, there is actually a lot of differentiation, so you’d better check out the two links above!

Integration requires a lot of logic and strategy; everytime you think: “oh, it would be nice if there was an x here” well, that might actually be what will help you solve the integral! Or if you see a “1” you could think of it as “sin^2(x)+cos^(2)” which is, indeed, 1. There are so many tricks and they just can’t be explained… you have to learn them by seeing the resolution. This is how I learnt to solve integrals, thanks to videos on Youtube. I highly recommend you the channel “blackpenredpen”. This guy is awesome!

Next lesson is going to be about the rules of integration, and after that you will be ready for integration techniques!

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