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Indefinite Integrals - Concept
Indefinite integrals are functions that do the opposite of what derivatives do. They represent taking the antiderivatives of functions. A formula useful for solving indefinite integrals is that the integral of x to the nth power is one divided by n+1 times x to the n+1 power, all plus a constant term.
Pretty soon we're going to start evaluating indefinite integrals but I want to lay down some ground work first. Let's recall that the expression, the integral of f of x dx is called an indefinite integral and what that represents are the antiderivatives of this function little f of x. And here is the relationship between a function and its antiderivatives, the antiderivative of capital F of x equals little f means that the integral of little f of x equals capital F of x plus c. So if you find one antiderivative and that's a function who's derivative is the given function, then you can add a plus c and that's all the antiderivatives of your given function. So we're going to be solving like this in the next few lessons.
Let's look at an example though I mean you could just use what you know about derivatives to create integral formulas for example the derivative of 5x squared minus 4x equals 10x-4 so you can turn that around. The integral of 10x-4 is 5x squared minus 4x+c just add a +c to your original function and that gives you all the antiderivatives of 10x-4. A useful integral formula we'll be using this a lot, this is how to integrate a power function x to the n the antiderivatives will be 1 over n+1 times x to the n+1+c and this formula only works if n is a negative 1 bringing another formula within is negative 1.
And there're 2 properties that we'll use a lot first the constant multiple rule, if you're integrating a function that has a constant in front of it, you can pull that constant out of the integral so that's going to be very useful. The other one is the sum rule, the integral of a sum is the sum of the integrals so if you have a sum or difference you can actually separate the integral over the sum which is also very useful. So we'll use these 2 properties and this formula in our coming lessons.