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In this unit, we'll discuss techniques for finding integrals, both definite and indefinite. The first technique, integration by substitution, is a way of thinking backwards. Then we'll directly apply formulas. Finally, we'll talk about improper integrals, which are integrals where either the limits of integration or the function contain something crazy, like an asymptote. Sans Shmoop, improper integrals can hurt the brain. As an example, these graphs don't look very different:
However, the area between and the x-axis on the interval (0,1] is finite, while the area between and the x-axis on the interval (0,1] is infinite!
A definite integral is a number. The quantity
is the weighted area between f and the horizontal axis as x moves from a to b.
An antiderivative is a function. The function
F ( x ) = x2
is an antiderivative of the function f ( x ) = 2x.
F2(x) = x2 + 2
is also an antiderivative of the function f ( x ) = 2x.
An indefinite integral is a family or collection of functions. To write an indefinite integral we use the integral sign without limits of integration. The indefinite integral
is the collection of ALL antiderivatives of the function f (x) = 2x. This means includes the following functions, and many others:
The collection makes infinitely many functions, so we can't name them all. We abbreviate the entire collection by writing
In the equation
the letter C is called the constant of integration. The function f (x) is still called the integrand, like it was for the definite integral.
The indefinite integral
and the general solution to the differential equation both describe the same thing: the collection of all functions with derivative f (x). Whether we're working in the context of integrals or of differential equations, we use the constant + C to describe all antiderivatives or all solutions at once.
One thing we haven't addressed yet is why two functions with the same derivative must be different by a constant (as opposed to different by something else). Suppose both F ( x ) and G ( x ) are antiderivatives of f ( x ). This means
F ' (x) = f (x)
G ' (x) = f (x).
Using our rules for taking derivatives,
This means the derivative of the function (F – G) is zero. Since only constants have derivative zero, (F – G) must be a constant number. If we start with one antiderivative F(x) of a function f(x), any other antiderivative G(x) must be F(x) plus some constant.
This explains why we can account for all antiderivatives of 2x by writing
x2 + C.
Evaluating an indefinite integral is the same thing as thinking backwards to find an antiderivative or finding the general solution to a differential equation. We find the simplest function whose derivative is what we want, and stick "+ C" on the end.
Techniques of integration are methods we can use to find antiderivatives and indefinite integrals. Since the FTC allows us to evaluate a definite integral using an antiderivative of the integrand, these techniques are also useful for finding definite integrals. We have to be extra-careful when evaluating definite integrals, since the limits of integration make things more complicated.
Problems that ask you to find indefinite integrals are nice, because you can always check your answers. For example, suppose the problem said
and we came up with the answer
x2ex – x2 + C
but we're not sure if that's right. Our answer is supposedly the family of all antiderivatives of the integrand
We take the derivative of our answer; if we get x2ex our answer is correct; if not, it isn't.
(x2ex – x2 + C)' = (x2ex + 2xex) – 2x + 0
That's not equal to x2ex, so we didn't have the right answer.
It's important to keep definite integrals and indefinite integrals straight.
A definite integral is an integral of the form
The integral sign has limits of integration. A definite integral is a number. There's no need to write + C.
An indefinite integral is an integral of the form
There are no limits of integration on the integral sign. This indefinite integral is the family of all antiderivatives of f (x). Remember to write + C!