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In this article we're going to be going over the basics of taking integrals (as you can tell from the title). This basically just means that we're going to discuss the most fundamental rules and tricks you'll need to know, but we won't be going over 'u-sub' and 'integration in parts'. Those two topics are a lot lengthier and take a lot more time to understand usually, we'll have a separate article for each coming out soon, so stay tuned!
Ok, let's jump right into it then!
So, what is an integral?
Most people define an integral as being the 'area under the curve' between the x-axis and the graph of a function.
The equation for an integral looks something like this
- 'a' and 'b' are the bounds for the integral. When an integral has the bounds, it's called a definite integral. If it doesn't have the bounds, then it's called an indefinite integral.
- f(x) is the function itself
- '∫' is what tells you that you're taking an integral
- The 'dx' is at the end of every integral. It not only represents the infinitesimal change in 'x' but also signals the end to the integral, so it's important you don't forget about adding it to the end of all your integrals.
This is especially important when you have an expression that has integrals and other types of functions.
Let's go over 'dx' a little more because it's a pretty important concept in calculus
- We could get the area under the curve by 'splitting up' the graph of a function and adding up the area of them like this
- We can see that the shorter the width of the rectangles, the closer and closer we get to the actual area under the curve. 'Δx', or the change in x, is the same as the width of the rectangles. Therefore, the smaller 'Δx' is, the closer we get to the actual area
- We refer to 'Δx' getting closer and closer to zero (making rectangles as thin as possible) as 'dx'
There are other ways to estimate the area under the curve too (riemann sums), but an integral is the most accurate.
Function-wise, an integral is the opposite of a derivative.
- In other words, if you took a derivative of a function, then the integral of the derivative would take you back to the original function.
Ok, now that we have that out of the way, we can get into the rules you'll need to know!
Just for these rules, we're going to being using indefinite integrals, or integrals with no bounds.
Reverse power rule
In the 'How to Take a Derivative' article, we discussed power rule (divide by the exponent, and then subtract one from the exponent). Reverse power rule is very similar to that.
- If we consider the exponent of the function as 'n', the first step in taking the integral would be to add one to it
- Next, you have to divide the whole term by 'n + 1'
- One thing to note, is that if you're taking an integral of a function that has a term with no variable, you would still have to use the reverse power rule on it!
This one's pretty straightforward, but we'll go through a quick example just in case. Say we were given this integral
- First, we need to identify the exponent or 'n'. In this case, it's '4'. The next step in taking the integral would be to add one to it, or find 'n + 1'.
- The last step is to divide the whole term by 'n + 1', which in this case is '5'.
- We very briefly went over the case in which you would be taking an integral of a function with no variable in it, and this is what it would look like.
- So, we see that our 'function' is 4. The exponent of this function is zero (since x^0 is one). This is super super important to remember, especially since it's so easy to forget. In other words this function could be written as
- Now, we can go through the steps of reverse power rule. Our exponent or 'n' is 0, so 'n + 1' is 1.
- And that's how to use the reverse power rule!
Multiplying by a constant
In most cases, the integral you're taking will have some type of a coefficient in front of it.
- The 'multiplying by a constant' rule makes the power rule (where you're dividing the whole term, including the constant), by another number
- In the case of our last example, we could've taken the integral a little differently using this rule as well. And it would look a little something like this
Overall, this rule just makes looking at the function you're working with a little easier, which is especially important when you have long complicated ones. But remember, if you have an integral that has multiple terms and you want to use this rule, you have to make sure that the constant you're 'taking out' is being taken out of all the terms.
Sum and Difference Rule
Like in the 'How to Take a Derivative' article, the sum and difference rule isn't all that confusing but it can definitely make your life easier.
This rule is basically just stating that if you're taking an indefinite integral or definite integral with multiple terms, you can split it up and take them separately.
If I could make 'plus c' a flashing neon sign and tape it to my forehead, I would. As you probably saw in the previous rules and examples, at the end of all the integrals, I put '+ c'.
- 'C' is formally known as the constant of integration. Fancy, I know. But, it really just means, constant.
- See, when we take a derivative, any constants become null because they don't have a variable associated with them.
- But, if we were to take the integral of '6x', just for the sake of this example, using reverse power rule we would have:
- The derivative of '3x^2 + 9012391924' would ALSO just be '6x', because 9012391924 is just a constant in the function
- We know that '3x^2' is not the same as '3x^2 + 5'. But the reverse power rule doesn't say anything about how to get to the exact derivative we started off with! Thats where '+c' comes into play :)
- So, when we take the integral of a function, we only know what the function is according to the terms that have a variable with them and nothing about the constants (if there are any). Therefore, to include this concept when taking integrals, make sure to put '+c' at the end of all of them!
Now you know how to take some integrals with these rules! Just remember to look out for negative exponents and fractional constants too!
📌 If you need some extra help with these concepts check out these awesome links!
- http://archives.math.utk.edu/visual.calculus/4/integrals.2/index.html - This one's could be considered bit of a challenge, just a heads up!
- Check out our 'Integrals to Memorize' article too!