Integration by Substitution Separable Differential Equations The chain
Integration by Substitution & Separable Differential Equations
The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for only a limited range of functions. We can sometimes use substitution to rewrite functions in a form that we can integrate.
Example 1: The variable of integration must match the variable in the expression. Don’t forget to substitute the value for u back into the problem!
Example: (Exploration 1 in the book) One of the clues that we look for is if we can find a function and its derivative in the integrand. The derivative of is Note that this only worked because of the 2 x in the original. Many integrals can not be done by substitution. .
Example 2: Solve for dx.
Example 3:
Example: (Not in book) We solve for because we can find it in the integrand.
Example 7:
Example 8: The technique is a little different for definite integrals. new limit We can find new limits, and then we don’t have to substitute back. We could have substituted back and used the original limits.
Example 8: Using the original limits: Leave the limits out until you substitute back. Wrong! The limits don’t match! This is usually more work than finding new limits
Example: (Exploration 2 in the book) Don’t forget to use the new limits.
Separable Differential Equations A separable differential equation can be expressed as the product of a function of x and a function of y. Example: Multiply both sides by dx and divide both sides by y 2 to separate the variables. (Assume y 2 is never zero. )
Separable Differential Equations A separable differential equation can be expressed as the product of a function of x and a function of y. Example: Combined constants of integration
Example 9 Separable differential equation This is a standard integral: =arctan(y) Combined constants of integration
Example 9: We now have y as an implicit function of x. We can find y as an explicit function of x by taking the tangent of both sides. Notice that we can not factor out the constant C, because the distributive property does not work with tangent.
In another generation or so, we might be able to use the calculator to find all integrals. Until then, remember that half the AP exam and half the nation’s college professors do not allow calculators. You must practice finding integrals by hand until you are good at it! p
Integration by Substitution Lesson 7
Substitution with Indefinite Integration • This is the “backwards” version of the chain rule • Recall … • Then …
Substitution with Indefinite Integration • In general we look at the f(x) and “split” it § into a g(u) and a du/dx • So that …
Substitution with Indefinite Integration • Note the parts of the integral from our example
Example • Try this … § what is the g(u)? § what is the du/dx? • We have a problem … Where is the 4 which we need?
Solution Substitute u=4 x− 5 �dx= Undo substitution u=4 x− 5: or or
Can You Tell? • Which one needs substitution for integration? • Go ahead and do the integration.
Try Another …
Assignment A
Change of Variables • We completely rewrite the integral in terms of u and du • Example: • So u = 2 x + 3 and du = 2 x dx • But we have an x in the integrand § So we solve for x in terms of u
Change of Variables • We end up with • It remains to distribute the and proceed with the integration • Do not forget to "un-substitute"
Change Of Variables 1. Example 1 Determine the new region that we get by applying the given transformation to the region R. a) R is the ellipse and the transformation is Solution There really isn’t too much to do with this one other than to plug the transformation into the equation for the ellipse and see what we get. So, we started out with an ellipse and after the transformation we had a disk of radius 2.
What About Definite Integrals • Consider a variation of integral from previous slide • One option is to change the limits § u = 3 t - 1 Then when t = 1, u = 2 when t = 2, u = 5 § Resulting integral
What About Definite Integrals • Also possible to "un-substitute" and use the original limits
Integration of Even & Odd Functions • Recall that for an even function § The function is symmetric about the y-axis • Thus • An odd function has § The function is symmetric about the orgin • Thus
Example Integrate by Substitution Steps
Usual Change of Variable
Usual Change of Variable 5. In the rational functions of radicals with different indices and the same linear radicand, ax + b, the change of variable is t raised to the least common multiple of the indices. 6. If 7. If is even: not even:
Assignment B
7. 1 Substitutions in Multiple Integrals Copyriht © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Definition of the Jacobian
Example 1 Find the Jacobian for the change of variables x = r cosө and y = r sinө
Example 1 Solution Find the Jacobian for the change of variables x = r cosө and y = r sinө
Why would we change variables?
Example 2 Let R be the region bounded by the lines x - 2 y = 0, x – 2 y = -4, x + y =4 and x + y = 1 Find a transformation T from region R to region S such that S is a rectangular region.
Example: 2 Solution
Example 2 Solution We can convert individual points between coordinate systems Similarly, we could use these formulas to convert in the other direction
Change of variables
Example 3 use a change of variables to simplify a region Let R be the region bounded by the lines x - 2 y = 0, x – 2 y = -4, x + y =4 and x + y = 1 as shown below. Evaluate the double integral.
Example 3 Solution slide 1
Example 3 Solution slide 2
Example 4 Let R be the region bounded by vertices (0, 1), (1, 2) (2, 1), (1, 0) a) Sketch the transformed region b) Evaluate the integral
Example 4 a Let u = x + y Let v = x- y
Example 4 solution Let u = x + y Let v = x- y
- Slides: 62