Welcome to another video
on the antiderivative and indefinite integration.
The main purpose of this second video
is to provide additional examples
involving trigonometric functions.
So, I would recommend that you watch
the previous video first,
because it will explain the concept
of antidifferentiation in much more detail,
but let's go ahead and do a quick review.
Let's say we knew that the derivative function
was equal to cosine x,
but we wanted to know what the original function
would have been.
I'm sure with some thought, you could figure out
that the original function must've been
f of x equals sine x,
but actually, there's an infinite number of functions
that have a derivative of cosine x.
We could add any constant on here.
For example, we could have f of x equals sine x plus one,
or plus two,
or any constant.
So, what we could do is say, f of x equals sine x plus c,
where c would be any constant.
And, we call this function the antiderivative
of f of x equals cosine x.
So, the reverse of differentiating is antidifferentiating,
and the result is called the antiderivative.
A function big f of x is the antiderivative of little f of x
if
the derivative of big f of x is equal to little f of x.
So, in the previous problem, we would've had f of x
equals cosine x,
and big f of x is equal to sine x plus c,
because a derivative of big f of x
is equal to small f of x.
And, we already said, to represent the entire family
of antiderivative, we can add a constant,
which we represent by plus c.
So, to formalize the process of antidifferentiation
is called integration, or indefinite integration.
To indicate the antiderivative of f of x equals cosine x
is equal to big f of x equals sine x plus c,
we write, the antiderivative or indefinite integral
of cosine x with respects to x,
equals sine x plus c.
This is called the integral sign.
This function here is called the integrand.
Dx is the variable of integration.
C is the constant of integration,
and big f of x plus c is our antiderivative.
Okay, so in previous video, we discussed
all of these new integration formulas,
and we compared them to
the corresponding differentiation formulas.
In this video, we want to focus on the integration formulas
for the trigonometric functions, as we see here.
You take a close look at these integration formulas,
and compare them to each differentiation formulas.
Hopefully they'll make sense.
Remember, if you take the derivative
of these antiderivatives,
we should end up with these integrands,
and we will if we apply these differentiation formulas.
Let's go ahead and try a few.
We want to determine the indefinite integral
of x cube minus sine x.
Well, to find the antiderivative of x cube,
we have to apply the power rule from the previous video.
So, we would add one to the exponent,
and then divide by four,
minus the antiderivative of sine x is equal to
negative cosine x,
and then plus c.
So, if we clean this up, this would be x to the fourth
divided by four, plus
cosine x plus c.
Notice we applied two different integration formulas.
Both had a plus c, but there's no reason
to put plus c twice, because if we add two constants,
we just get another constant.
So, it's common practice just to put one plus c
at the end of our antiderivative.
Number two, we have the indefinite integral
of two secant squared x plus cosine x,
which would equal two times the antiderivative
of secant squared x,
which is equal to tangent x plus c,
plus the antiderivative of cosine x
is equal to sine x
plus c, and again, we just put our plus c on the end.
Let's go and take a look at two more
that aren't quite so straightforward.
Sometimes, we have to manipulate the integrand
in order to make it fit one of these integration formulas.
So, for example, on this problem,
we need to eliminate the parentheses first,
so we'll distribute the secant x.
So, we'll have the indefinite integral of secant x tangent x
minus secant squared x,
all with respects to x.
Okay, now, if we take a look at our formulas,
we do have two that fit.
The antiderivative of secant x times tangent x
is equal to secant x minus the antiderivative
of secant squared x is equal to tangent x,
and then, of course, we have the plus c.
Okay, on this last example,
again, we notice right away, it doesn't fit one of our
basic integration formulas, so we have to manipulate this.
One thing that comes to mind on this problem,
one minus sine squared x is equal to cosine squared x.
So, let's go ahead and try to replace
one minus sine squared x with cosine squared x.
Next, we could rewrite this as the indefinite integral of
one over
cosine x
times
sine x
over cosine x.
And, you might ask, "Well, why would you want to do that?"
well, again, the more familiar you become
with these formulas, the more tricks you'll learn,
but one over cosine x is equal to secant x,
and sine x divided by cosine x is equal to tangent x.
And now, if you take a look at the formulas,
we can go ahead and use this one again,
where we have the antiderivative of secant x tangent x
is equal to secant x plus c.
Okay, I hope you found these additional examples helpful.
Thank you for watching.