We've all done it. Anyone who's taken calculus in their life has at some point forgotten to put the at the end of an indefinite integral. But there's more to the then just something you have to remember or you'll lose points on your calculus test. I'm going to discuss a couple integrals and explain how the can help you avoid mathematical disaster.
First, an explanation of the What is its purpose? When you take an indefinite integral of you are figuring out what function has the derivative equal to Take a look at two functions, and Obviously, these are different functions, but when you derive them, you will find that So if we are trying to find the antiderivative of we add a meaning adding a constant to the antiderivative, to account for the fact that derives to for all real
So let's take a look at a couple integrals.
This first one has several different ways to find the antiderivative.
Let's start with my preferred way.
But we can make a couple different too. Let and
Now let and
Somehow, these are all equal to each other. But how? Let's take a look at some properties of before coming back to these integrals.
Remember that is a constant between and Don't look at as a number, but a function of with a range of So we can say, for example, this.
As long as the function of has range then you can use it, but you may need to restrict the domain. Take a look at this.
One last thing to note before looking at the integral again is that not all are the same. Subtraction of these integrals does not take away the Here is the misconception.
But those are A constant minus a constant is another constant.
This makes sense because of this.
So we've found some properties of the constants in integrals. Let's go back to the integral of
We have already proved that Let's find the constant that will make
Expanding the LHS,
Look at that! Cancel out the to find that so The difference between the constants of the two integrals is
So let's conclude our proof that by saying this. Here, the are equal to each other.
Not all integrals in need of recalibrating the involve manipulating multiple The next one we will work with is this.
But that integral looks like it uses hyperbolic geometry. In fact, it is simply this.
We have another case of needing to manipulate the
Using the properties of logarithms, But is a constant. So and the integrals are equal!
So now we've seen how we can manipulate the in an indefinite integral to make sure that the laws of math aren't broken. Here are some strategies for manipulating the constant.
Separate the with subscripts. Not all constants are the same.
The difference between two constants is still a constant that may or may not be
represents a function with range
Often, you can use properties of logarithms (or other functions) to separate out constants from a variable expression.
If the simplified version of your indefinite integral contains an integer added to a variable expression, you may want to add the integer to to produce a new
Discuss this integral with respect to manipulating the constants.
Can you share a problem that uses this type of variable manipulation? Do you have any comments about this topic? Please comment and leave your ideas.
Thanks for reading this post, and I hope it helps out! I'll be using the hashtag #TrevorsTips for posts that talk about seemingly simple concepts that can actually be really helpful.