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# 6.2 indefinite integrals.pdf

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University of Toronto St. George

Mathematics

MAT237Y1

Dan Dolderman

Fall

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Calc 1
Fri 11/7/08
6.2 - The Inde▯nite Integral
In 6.1, we were basically calling the antiderivative A(x). Well, that was mostly to get you to see
that it means the area function under the curve f(x). We usually denote the antiderivative of f(x)
as F(x) (just like how we denoted the derivative of f(x) to be f (x)...same thing). So, if F(x) is
the antiderivative of f(x), then that means
F (x) = f(x):
Remember...integral...antiderivative...area under the curve. They all mean the same thing.
Example: F(x) = x is an antiderivative of f(x) = x. Why? Take the derivative of F (x) to
2
▯nd out.
d 1 2 1 2
dx 2x = 2▯ 2x = x
Notice how I said an antiderivative, not the antiderivative. Remember from 6.1 how
antiderivatives are not unique at all! There can be that constant at the end. I can’t stress it
enough to remember that +C after each time you ▯nd the antiderivative.
Time to introduce new notation. Just like when we were dealing with derivatives, we created a
symbol to mean \take the derivative of this function" and it was f(x), there is a special symbol
Z dx
for integrals (antiderivatives). This symbol if(x) dx. That funky S-looking thing means
\integral," f(x) is the function you want to take the integral of, and \dx" means you want to take
the integral with respect to x (just like the dx in the denominator of the derivative).
Z
1 2
x dx = x + C
2
This is called the Inde▯nite Integral. ...As opposed to the De▯nite Integral, which would be over
an interval (like, say, [▯2;4]. This will come later). Inde▯nite integrals always give out a function,
and so you will always need that +C.
Since integration is the opposite of derivation, this means that
d ▯Z ▯
f(x) dx = f(x)
dx
if you take the derivative of the integral, you get back the function that you started with. Same
think if you take the integral of the derivative. It’s kind of like when you take a function and
compose it with the inverse of that function, you get back the variable you started with.
Example:
Z
d
x2 = 2x 2x dx = x + C
dx Z
d 1 1
x4 = 2x 3 2x dx = x + C
dx 2 Z 2
d
3x = 3 3 dx = 3x + C
dx
d 1 5 Z 5 1
x5 = x4 x dx = x + C
dx 3 3 3 3
1 d n n▯1
Notice any patterns here? Remember the power rule for derivativedxx = nx . Any guesses
as to what the power rule for antiderivatives may be?
Z
n xn+1
x dx = + C
n + 1
(just make sure that n 6= ▯1. That’s its own special case. Care to guess what happens when
n = ▯1?)
Soooo, Z Z
1 dx = dx?
Z
dx = x + C
Z
x dx?
Z
1 x2
x dx = x + C = + C
2 2
Z 2
x dx?
2
Z
x2 1 3 x 3
dx = x + C = + C
2 6 6
So, just like derivatives, we can just keep on taking the integral of a function. In fact, because
you’re working backwards, sometimes you can end up with a nicer function after ▯nding the
antiderivative! This is because being di▯erentiable was stronger than being continuous. So, while
d jxj gave you that nasty jump discontinuity, you can actually start with that same function with
dx
the jump discontinuity and after ▯nding its antiderivative, you end up back with jxj + C!
Now let’s take a look at polynomials:
Z
x + 2x ▯ 4x + 5 dx
x4 2x3 2
(= + ▯ 2x + 5x + C)
4 3
Or perhaps this guy: Z
3x▯4 ▯ x▯2+ 5x dx
(= ▯x ▯3 + x▯1 + 5x + C)
3
Remember: the idea behind integrals is that you have to think backwards. Think, \okay, the
derivative of what gives me this function?" Here are some trig examples to ponder:
Z
cos(x) dx
(= sin(x) + C)
2 Z
sin(x) dx
(= ▯cos(x) + C)
Z
sec (x) dx
(= tan(x) + C)
Z
sec(x)tan(x) dx
(= sec(x) + C)
d x x
That’s enough trig stu▯ for now. Let’s go onRto other examples. Remember how dxe = e ? Sure,
how could you forget? So, what do you think e dx is? e + C!
1
So what happens when you have x (remember how i said you can use the power rule backwards
for everything but x1 )? Well, think about it for a second. The derivative of which function gives
1
you x ? lnjxj! This means that Z
1
dx = lnjxj + C

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