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# Rewriting before integrating: challenge problem

In this example, we find the antiderivative an expression which is not so simple.  Created by Sal Khan.

## Want to join the conversation?

• Correct me if I'm wrong, but when you apply the sum rule to break the antiderivative into smaller pieces, you would technically have an added constant for each piece. At when you add the constant, that constant would technically represent the sum of all smaller constants found when taking the general antiderivative of each of the smaller pieces, correct? It would be almost impossible to find a particular solution given an initial condition, since it we would be solving for multiple constant C's.
• c can be anything, as opposed to x, which refers to a single, unknown value. We add c to generalise the (INDEFINITE) integral, so c can be anything, and everything, it doesn't matter!!
c = 0, 1, 2, 3, 4...(to infinity) and all the values in between, and all the negative numbers and and and...(you get the idea...basically all the numbers in the universe - can someone please let me know if this includes complex numbers...?)

To conclude,, any number, plus any number is still, any number! (c + c = c)

I emphasised INDEFINITE, because definite integrals refer to the area under the curve/line. (gross oversimplification!!) Clearly, changing c (or the height) would change the area. So c for definite integrals can only be one, specific, value, if any..
• why is it necessary to put that dx after every integral sign. Whats the purpose?
• I would disagree that the dx is mostly for show.

You can think of integration simply as the sum of a lot of really skinny (infinitely skinny, but not zero) rectangles. Now, how do you get the area of a rectangle? Well, you multiply its length times its width. Integration just takes the length times the width of a bunch of skinny rectangles and sums them. So how does that connect to the integration expression?

Let's say you have an integral that looks like this: ∫x^2dx. There are three parts to the integral: 1) the function part, x^2, represents the length of the rectangles--you can also think of it as the height of the function, or the y-value; 2) the dx in the integral represents the width of each rectangle; 3) and the elegant elongated S-symbol, ∫, tells you to sum the areas of the rectangles.

Watch Sal's videos on Riemann sums and it will become obvious what dx means and why you need to have it there. Without the dx, you can't have any width. And if you don't have any width, then you can't have any area. And no area means nothing to sum. And nothing to sum means no integration. And no integration means no differentiation (you get a glimpse of the link between the two in this video, but you will see it in a profoundly beautiful way in the Fundamental Theorem of Calculus) . And no differentiation means no calculus. And think of how horrible the world would be without calculus. :)
• Why are these called indefinite integrals?
• Because the Integral doesn't have any Boundaries it needs to relate to.
• Is there a way to solve for C in the equation, or for any antiderivative?
• If you've given any initial conditions, then yes. If not, then not.
For example, suppose you're given that f´(x) = 2x and f(0) = 1. Taking the antiderivative of f´(x) gives you x^2 + C, and with the initial condition that f(0) = 1 you can solve for f(x) = x^2 + 1.
• Holy cow, I was sitting here wondering what a "hairier expression" is. I was thinking like a Fourier or something. I get it now.
• Hi Sal,
Upto my understanding Integration is the inverse of Differentiation. So if we integrate 2x, we will get x^2. It means that d/dx (x^2) is 2x. My question is why should we neglect constants in integration? why not we can integrate some constant, say integration of 8 dx and it should be 8x accordingly . Because differentiation of 8x is 8 (a constant) right? Hope you will understand my doubt. Please reply me sal
thanks for your nobel service sal
Srini
• You are correct that if we are told to find the indefinite integral of 8, it would be 8x (because the derivative of 8x is 8). But if you are told to find the derivative of x^2 + 8, your answer is simply 2x. What about finding the derivative of x^2 + 9? Again, your answer is 2x. So, some information about the original function is lost when looking only at its derivative. This is the reason why we write "+ C" when finding the anti-derivative, because the derivative of x^2 + any constant is 2x. If there is no constant in the original function, you could still think of it as x^2 + 0.
• I would really appreciate to look at the graph of such kind of an indefinite integral as to where do you actually put that constant C in that graph, i want to see it located in that particular graph cuz in definite integrals we have area under that specific part of that curve according to the limits we put up Its really confusing about the representation of C in the graph of indefinite integrals :-0
• Graphically the integrals gives you the a "family of "parallel' curves" not a unique curve... that's what the "C" is doing here
• Hi Sir! I truly admire the way you teach, but I think their is something in this video of hairy integration that had been overlooked by Sir Khan. Please correct me if I'm wrong cause I'm just starting to learn this subject. The thing you had overlooked is in the 3rd term. the exponent you had written is -3/2 instead of -5/2. thank you :D
• Lets take the anti-derivative of the third term alone, but first, some algebraic manipulation:

(18x^(1/2))/(x^3)=18x^(1/2-3)=18x^(1/2-6/2)=18x^(-5/2)

Now we take the anti derivative of 18x^(-5/2):

Here we raise the exponent with 1 power:

-5/2+1=-5/2+2/2=-3/2

And divide by the exponent:

[18x^(-3/2)]/[-3/2]=-12x^(-3/2) (+c if you will)

For negative constants of exponents it might seem confusing to raise a power by one, but this is how we would do it.

I hope this was a little helpful!