1. WARMUP 1)

2. 2)

3. 6.2:Integration by Substitution
& Separable Differential Equations

4. 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.

5. 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!

6. Note that this only worked because of the 2x in the original.
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 2x in the original.
Many integrals can not be done by substitution.

7. Example 2:
We can’t replace dx with du, since
they are not equal
Multiply by 4 to get them equal,
but to “balance it”, multiply the
outside by the reciprocal.

8. Example 3:

9. Example: (Not in book)

10. Example 7:

11. The technique is a little different for definite integrals.
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.
new limit
We could have substituted back and used the original limits.

12. Wrong! The limits don’t match! 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

13. Example: (Exploration 2 in the book)
Don’t forget to use the new limits.

14. 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 y2 to separate the variables. (Assume y2 is never zero.)

15. 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

16. Example 9:
Separable differential equation
Combined constants of integration

17. 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.

18. the end