In this video we are going to talk about Trigonometric Substitutions. There are three main equations that we have for trigonometric substitution. First we are going to list them. The idea is that if we see any one of these three forms in an integral, then we can do what is called a Trigonometric Substitution to convert out integral into a trigonometric integral. Then we can use some of the trigonometric techniques that we learned in the previous lecture to solve the integral. The best way to learn this is to do lots of examples, so we are going to do some of them.
Remember to make the
substitution as well as the main substitution.
can sometimes use trigonometric substitution even when you dont
have a square root.
However, in these cases, you should only be using the tangent
substitution. If you get one of the other ones, then you should have
been able to factor the algebraic expression, and trigonometric
substitution was not necessary.
If you have a linear term (x or u, as opposed to x² or
u²), complete the square and make a substitution to eliminate the
linear term before you make the trigonometric substitution.
If the coefficient of the quadratic term (x²) is negative,
factor the negative out of all terms before completing the square.
you make the trigonometric substitution, it becomes a trigonometric
integral that you can handle using the techniques in the previous
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If you have the square root of that expression, then that will convert into cos(θ), and you will get an easier integral to deal with.0096
By the same idea, if you have au2-b, then you are going to have u=sqrt(b/(asec(θ)).0104
What you are trying to do there is take advantage of the trigonometric identity tan2(θ)+1=sec2(θ).0117
Sec2(θ)-1 = tan2(θ) so once you make this substitution you are going to end up with sec2(θ)-1 under the square root.0131
The last one you will have a+bu2=sqrt(a/(btan(θ))) and so again you are taking advantage of this trigonometric identity tan2(θ)+1.0144
You will end up with that under the square root, and that will convert into sec2(θ).0161
All of these take advantage of these trigonometric identities.0165
One thing that is important to remember when you are making these substitutions is whenever you substitute u equals something, or x equals something, you always have to substitute in dU or dX as well.0170
For example, if you substitute u=sqrt(a/bsin(θ)), then you also have to substitute dU, which would be, a and b are constants so that is just sqrt(a/bcos(θ)) dθ.0182
You always have to make the accompanying substitution for your dU or your dX.0204
As with all mathematical problems it is a little hard to understand when you are just looking at mathematical formulas in general, but we will move onto examples and you will see how these work.0211
The first example is the integral of 4 - 9x2 dx.0223
That example matches the first pattern that we saw before, which was the square root of a minus b u2.0229
The substitution that we learned for that is u equals the square root of a/b sin(θ).0237
Here, the a is 4, the b is 9, the x is taking the place of the u.0249
What we are going to substitute is x equals well the square root of a over b, is the square root of 4/9 sin(θ).0259