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Integral Calculus (2017 edition)
Course: Integral Calculus (2017 edition) > Unit 6
Lesson 6: Trigonometric substitution- Introduction to trigonometric substitution
- Substitution with x=sin(theta)
- More trig sub practice
- Trig and u substitution together (part 1)
- Trig and u substitution together (part 2)
- Trig substitution with tangent
- More trig substitution with tangent
- Long trig sub problem
- Trigonometric substitution
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More trig sub practice
Example of using trig substitution to solve an indefinite integral. Created by Sal Khan.
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Why is the integral of d(theta) = theta?(27 votes)
All d(theta) means is that you are integrating with respect to theta. It's just like how dx means to integrate with respect to 'x'. An indefinite integral of (1/sqrt[2])dx would be (x/sqrt[2]). So if we took the integral of (1/sqrt[2]) with respect to theta, the answer is just (1/sqrt[2])*theta. Don't forget that (1/sqrt[2]) is a constant. To take the antiderivative of any constant, you just tack on the variable that you are dealing with.(53 votes)
- Someone help me out with the final answer! Arcsine has domain restrictions, how is this accounted for? do you need to simplify it further for a more complete answer? does this mean the value which arcsine is being evaluated at needs to fit within the domain, therefore x needs to be within the domain?(16 votes)
I had the same question and I didn't feel satisfied until I proved it to myself :) Here's the proof I wrote for peter.s.vdm's answer.
https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_squared_minus_x_squared(20 votes)
If I substitute sin^2theta = cos^2theta I get a negative sign in the final result. How can two answers be possible or am I making some mistake?(5 votes)- it's because arcsinθ = -arccosθ + π/2
See how important the + C is? :)(18 votes)
does dx mean a very small change in x?(4 votes)
At2:17, when he takes the square root of the equation, should he not put in a plus or minus (+-), since squaring a positive or a negative will give you the same result?(5 votes)
It would've gave the same answer in the end.(4 votes)
- When you make the substitution x=a*sin(theta), you are limiting the domain of x from all real numbers to the interval [-a, a]. Wouldn't this mean the answer is only valid for x in the interval [-a, a]? Also, how can we assume that sqrt(sin^2 x) = sin x and not -sin x?(3 votes)
- EXCELLENT QUESTION!
I am pretty sure Sal deals with these nuances of trig sub in other videos.
We can make the trig substitution x = a sin θ provided that it defines a one-to-one function. This can be accomplished by restricting θ to lie in the interval [-π/2, π/2] (for cos and sin).
The point of trig sub is to get rid of a square root, which by its very nature also has a domain restriction. If we change the variable from x to θ by the substitution x = a sin θ, then we can use the the trig identity 1 - sin²θ = cos²θ which allows us to get rid of the square root sign, since:
√(a² - x²) = √(a² - a²sin²θ) = √(a²(1 - sin²θ)) = √(a²cos²θ) = a|cosθ|
And since θ is in [-π/2, π/2], we can say that a|cosθ| = acosθ.(8 votes)
how do you solve for theta like sal did in2:46?(2 votes)- inverse trigonometric functions
sin (arcsin(x))=x
cos (arccos(x))=x
tan(arctan(x))=x
csc(arccsc(x))=x
sec(arcsec(x))=x
cot(arccot(x))=x(6 votes)
how the heck do you factor out 3 from 3-2x^2? first time i heard of this......(2 votes)- It is a great algebraic tool and a very useful one.
If you have 3 - 2x² that can be rewritten as 3(1- ⅔x), since when you multiply it out you get:
3(1) - (3)(⅔)x. You can see how the 3 cancels with the 3 in the denominator of ⅔, so that you get the original expression back - thus the expressions are equivalent.
You will see insights and algebraic manipulations like this more and more, espcially in integral calculus as we often need to transform an expression into another equivalent expression that we can find the integral for, just like was done in the video.(5 votes)
I'm having trouble understanding the trigonometric substitutions.
By substituting a x=sin(a) , aren't we limiting the values that x can take? Like sin(a) can only be a value between -1 to 1. So we are limiting the values of x to also be between -1 and 1. Isn't that saying we can not take any value of x other than those between these limits? And if so how is the substitution useful in the general case of solving a problem?(3 votes)
That's a very good question, and lots of people wonder the same thing.
If you look at the problem Sal is solving in this video, you'll see that the values of x already are limited. In particular, x has to be between ±√(3/2). If it's beyond that, then the part under the radical in the original equation will be negative.
When Sal finishes, you'll notice that the part he ends up with has the exact same limitation: x still only can be between ±√(3/2). If it's beyond that, then you'd leave the domain of arcsin.
Take a look at the other videos in this section (on trig substitution), and you'll notice that they all have similar limits from the outset.(2 votes)
At0:58he says you can use either sin^2 or cos^2, but won't that result in arccos at7:05instead of arcsin, which is a different answer? Or am I making a mistake?(2 votes)- I didn't watch the video, but arccos and arcsin differ by a constant and a negative sign only. So you'll end up with, say, -arccos instead of arcsin, and the constant difference will be absorbed by your arbitrary constant. Both are equally fine antiderivatives.(3 votes)
2:17Video transcript
Let's say I have the indefinite
integral 1 over the square root of 3 minus 2x squared. Of course I have a dx there. So right when I look at
that, there's no obvious traditional method of
taking this antiderivative. I don't have the derivative of
this sitting someplace else in the integral, so I can't do
traditional u-substitution. But what I can do is I could
say, well, this almost looks like some trig identities that
I'm familiar with, so maybe I can substitute with
trig functions. So let's see if I can find
a trig identity that looks similar to this. Well, our most basic
trigonometric identity-- this comes from the unit circle
definition-- is that the sine squared of theta plus
the cosine squared of theta is equal to 1. And then if we subtract cosine
squared of theta from both sides, we get-- or if we
subtract sine squared of theta from both sides, we could do
either-- we could get cosine squared of theta is equal to 1
minus sine squared of theta. We could do either way. But this, all of a sudden, this
thing right here, starts to look a little bit like this. Maybe I can do a little bit of
algebraic manipulation to make this look a lot like that. So the first thing, I would
like to have a 1 here-- at least, that's how my brain
works-- so let's factor out a 3 out of this denominator. So this is the same thing as
the integral of 1 over the square root of-- let me factor
out a 3 out of this expression. 3 times 1 minus 2/3x squared. I did nothing fancy here. I just factored the 3
out of this expression, that's all I did. But the neat thing now is,
this expression looks a lot like that expression. In fact, if I substitute, if I
say that this thing right here, this 2/3x squared, if I set it
equal to sine squared theta, I will be able to use
this identity. So let's do that. Let's set 2/3x squared, let's
set that equal to sine squared of theta. So if we take the square root
of both sides of this equation, I get the square root of 2 over
the square root of 3 times x is equal to the sine of theta. If I want to solve for
x, what do I get? And, well, we're going to
have to solve for both x and for theta, so
let's do it both ways. First, let's solve for theta. If we solve for theta, you get
that theta is equal to the arcsine, or the inverse sine,
of square root of 2 over square root of 3x. That's if you solve for theta. Now, if you solve for x, you
just multiply both sides of this equation times the inverse
of this and you get x is equal to-- divide both sides of the
equation by this or multiply it by the inverse-- is equal to
the square root of 3 over the square root of 2 times
the sine of theta. And we were going to substitute
this with sine squared of theta, but we can't leave
this dx out there. We have to take the integral
with respect to d theta. So what's dx with
respect to d theta? So the derivative of x with
respect to theta is equal to square root of 3 over
square root of 2. Derivative of this with respect
to theta is just cosine of theta, and if we want to write
this in terms of dx, we could just write that dx is equal to
square root of 3 over the square root of 2 cosine
of theta d theta. Now we're ready to substitute. So we can rewrite this
expression up here-- I'll do it in this reddish color-- I was
using that, let me do it in the blue color. We can rewrite this
expression up here now. It's an indefinite integral
of-- dx is on the numerator, right? Instead of writing this 1
times dx, I could have just written a dx up here. That could be a dx
just like that. You're just multiplying
it times dx. So what's dx? dx is this business. I'll do it in yellow. dx is this right here. So it's the square root of
3 over the square root of 2 cosine theta d theta. That's what dx was. Now, the denominator in my
equation, I have the square root of 3 times--
now it's 1 minus. Now I said 2/3x squared is
equal to sine squared of theta. Now how can I simplify this? Well, what's 1 minus
sine squared of theta? That's cosine squared of theta. So this thing right here is
cosine squared of theta. So my indefinite integral
becomes the square root of 3 over the square root of 2
cosine theta d theta, all of that over the square root of
3 times the cosine squared of theta. That just became cosine
squared of theta. So let's just take the square
root of this bottom part. So this is going to be equal
to-- I'll do an arbitrary change of colors-- square root
of 3 over the square root of 2 cosine of theta d theta, all of
that over-- what's the square root of this? It's equal to the square root
of 3 times the square root of cosine squared, so
times cosine of theta. Now, this simplifies
things a good bit. I have a cosine of theta
divided by a cosine of theta, those cancel out, so we'll just
get 1, and then I have a square root of 3 up here divided by a
square root of 3, so those two guys are going to cancel out,
so my integral simplifies nicely to 1 over square
root of 2 d theta. Or even better, I could write
this-- this is just a constant term, I could take it out of my
integral-- it equals 1 over the square root of 2 times my
integral of just d theta. And this is super easy. This is equal to 1 over
the square root of 2 times theta plus c. Plus some constant. I mean, you could say that the
integral of this is theta plus c and then you'd multiply the
constant times this, but it's still going to be some
arbitrary constant. I think you know how to take
the antiderivative of this. But are we done? Well, no. We want to know our indefinite
integral in terms of x. So now we have to
reverse substitute. So what is theta? We figured that out here. theta
is equal to arcsine the square root of 2 over the
square root of 3x. So our original indefinite
integral, which was all of this silliness up here, now that I
reverse substitute for theta or put x back in there, it's 1
over the square root of 2 times theta. theta is just this, is just
arcsine of square root of 2 over square root of 3 x, and
then I have this constant out here, plus c. So this right here is
the antiderivative of 1 over the square root
of 3 minus 2x squared. So hopefully you
found that helpful. I'm going to do a couple of
more videos where we go through a bunch of these examples,
just so that you get familiar with them.